# OEIS A181980: Least positive integer m > 1 such that 1 – m^k + m^(2k) – m^(3k) + m^(4k) is prime, where k = A003592(n)

Posted in categories: Fun Stuffs, Prime Search

The sequence:

2, 4, 2, 6, 2, 20, 20, 26, 25, 10,

14, 5, 373, 4, 65, 232, 56, 2, 521, 911,

1156, 1619, 647, 511, 34, 2336, 2123, 1274, 2866, 951,

2199, 1353, 4965, 7396, 13513, 3692, 14103, 32275, 2257, 86,

3928, 2779, 18781, 85835, 820, 16647, 2468, 26677, 1172, 38361,

40842,

1-m^k+m^(2*k)-m^(3^k)+m^(4*k) equals Phi(10*k,m), or Phi(10, m^k).

Proofing status:
a(1)=Phi[10,2^1]: 2 digits, p^2-1 factored prime.
a(2)=Phi[10,4^2]: 5 digits, p^2-1 factored prime.
a(3)=Phi[10,2^4]: 5 digits, Equals to a(2)
a(4)=Phi[10,6^5]: 16 digits, p^2-1 factored prime.
a(5)=Phi[10,2^8]: 10 digits, p^2-1 factored prime.
a(6)=Phi[10,20^10]: 53 digits, OpenPFGW 129.07% proof without helper factorization.
a(7)=Phi[10,20^16]: 84 digits, OpenPFGW 131.88% proof without helper factorization.
a(8)=Phi[10,26^20]: 114 digits, OpenPFGW 143.88% proof without helper factorization.
a(9)=Phi[10,25^25]: 140 digits, OpenPFGW 131.03% proof without helper factorization.
a(10)=Phi[10,10^32]: 128 digits, OpenPFGW 124.71% proof without helper factorization.
a(11)=Phi[10,14^40]: 184 digits, OpenPFGW 142.53% proof without helper factorization.
a(12)=Phi[10,5^50]: 140 digits, Equals to a(9).
a(13)=Phi[10,373^64]: 494 digits, OpenPFGW 162.51% proof with factored part 53.73%.
a(14)=Phi[10,4^80]: 193 digits, OpenPFGW 301.25% proof with factored part 100.00%.
a(15)=Phi[10,65^100]: 544 digits, OpenPFGW 182.06% proof with factored part 60.38%.
a(16)=Phi[10,232^125]: 888 digits, OpenPFGW with factored part 41.45% and helper 0.56% (124.92% proof).
a(17)=Phi[10,56^128]: 896 digits, OpenPFGW with factored part 38.45% and helper 0.50% (115.94% proof).
a(18)=Phi[10,2^160]: 193 digits, Equals to a(14).
a(19)=Phi[10,521^200]: 2,174 digits, OpenPFGW with factored part 44.38% and helper 0.54% (133.68% proof).
a(20)=Phi[10,911^250]: 2,960 digits, OpenPFGW with factored part 33.79% and helper 0.01% (101.38% proof).
a(21)=Phi[10,1156^256]: 3,137 digits, CHG proof with factored part 28.90%.
a(22)=Phi[10,1619^320]: 4,108 digits, OpenPFGW with factored part 33.29% and helper 0.14% (100.04% proof).
a(23)=Phi[10,647^400]: 4,498 digits, OpenPFGW with factored part 38.43% and helper 0.15% (115.44% proof).
a(24)=Phi[10,511^500]: 5,417 digits, OpenPFGW with factored part 33.70% and helper 0.07% (101.17% proof).
a(25)=Phi[10,34^512]: 3,137 digits, Equals to a(21)
a(26)=Phi[10,2336^625]: 8,422 digits, kp proof with factored part 30.78%.
a(27)=Phi[10,2123^640]: 8,517 digits, OpenPFGW with factored part 40.68% and helper 0.06% (122.09% proof).
a(28)=Phi[10,1274^800]: 9,937 digits, kp proof with factored part 31.61%.
a(29)=Phi[10,2866^1000]: 13,830 digits, CHG proof with factored part 27.77%.
a(30)=Phi[10,951^1024]: 12,199 digits, CHG proof with factored part 26.95%.
a(31)=Phi[10,2199^1250]: 16,712 digit, CHG proof with factored part 27.09%.
a(32)=Phi[10,1353^1280]: 16,033 digits, CHG proof with factored part 28.05%.
a(33)=Phi[10,4965^1600]: 23,654 digits, CHG proof with factored part 29.10%.
a(34)=Phi[10,7396^2000]: 30,952 digits, CHG proof with factored part 28.58%.
*a(35)=Phi[10,13513^2048]: 33,840 digits, PRP with factored part 26.13%.
a(36)=Phi[10,3692^2500]: 35,673 digits, CHG proof with factored part 27.69%.
a(37)=Phi[10,14103^2560]: 42,489 digits, CHG proof factored part 26.40% (chgcertd Type error).
*a(38)=Phi[10,32275^3123]: 56,361 digits, PRP with factored part 25.84%.
a(39)=Phi[10,2257^3200]: 42,926 digits, CHG proof with factored part 27.90%.
a(40)=Phi[10,86^4000]: 30,952 digits, kp proof with factored part 30.02%.
a(41)=Phi[10,3928^4096]: 58,887 digits, CHG proof factored part 27.04%.
a(42)=Phi[10,2779^5000]: 68,878 digits, CHG proof with factored part 26.69%.
*a(43)=Phi[10,18781^5120]: 87,526 digits, PRP with factored part 26.34%.
*a(44)=Phi[10,85835^6250]: 123,342 digits, PRP with factored part 25.84%.
a(45)=Phi[10,820^6400]: 74,594 digits, CHG proof with factored part 27.12%.
*a(46)=Phi[10,16647^8000]: 135,083 digits, PRP with factored part 26.07%.
*a(47)=Phi[10,2468^8192]: 111,161 digits, PRP with factored part 25.28%.
a(48)=Phi[10,26677^10000]: 177,046 digits, CHG proof with factored part 27.16%.
*a(49)=Phi[10,1172^10240]: 125,704 digits, PRP with factored part 25.45%.
*a(50)=Phi[10,38361^12500]: 229,195 digits, PRP with factored part 25.32%.
a(51)=Phi[10,40842^12800]: 236,089 digits, CHG proof with factored part 28.14%.

Note: Phi[10,4^2]=Phi[10,2^4] since 4=2^2
Note: Phi[10,25^25]=Phi[10,5^50] since 25=5^2
Note: Phi[10,4^80]=Phi[10,2^160] since 4=2^2
Note: Phi[10,1156^256]=Phi[10,34^512] since 1156=34^2
Note: Phi[10,7396^2000]=Phi[10,86^4000] since 7396=86^2

a(1)=Phi[10, 2^1]=11

a(2)=Phi[10,4^2]=61681

a(3)=Phi[10,2^4]=61681

a(4)=Phi[10,6^5]=3655688315536801

OpenPFGW proof:

$./pfgw -tc -q”1-6^5+6^(2*5)-6^(3*5)+6^(4*5)” Primality testing 1-6^5+6^(2*5)-6^(3*5)+6^(4*5) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 7 Running N-1 test using base 13 Running N-1 test using base 17 Running N+1 test using discriminant 23, base 2+sqrt(23) Running N+1 test using discriminant 23, base 3+sqrt(23) Calling N+1 BLS with factored part 100.00% and helper 100.00% (403.92% proof) 1-6^5+6^(2*5)-6^(3*5)+6^(4*5) is prime! (0.0193s+0.0004s) a(5)=Phi[10,2^8]=4278255361 a(6)=Phi[10,20^10]=10995116277758926258176000104857599999989760000000001 OpenPFGW proof:$ ./pfgw -tc -q”1-20^10+20^(2*10)-20^(3*10)+20^(4*10)”
PFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14]

Primality testing 1-20^10+20^(2*10)-20^(3*10)+20^(4*10) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N-1 test using base 23
Running N-1 test using base 37
Running N+1 test using discriminant 43, base 3+sqrt(43)
Calling N-1 BLS with factored part 40.12% and helper 8.72% (129.07% proof)
1-20^10+20^(2*10)-20^(3*10)+20^(4*10) is prime! (0.0086s+0.0006s)

a(7)=Phi[10,20^16]=18446744073709551615971852502328934400000\
0429496729599999999999344640000000000000001

OpenPFGW proof:

$./pfgw -tc -q”1-20^16+20^(2*16)-20^(3*16)+20^(4*16)” PFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14] Primality testing 1-20^16+20^(2*16)-20^(3*16)+20^(4*16) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 29 Running N+1 test using discriminant 37, base 15+sqrt(37) Calling N-1 BLS with factored part 42.03% and helper 5.07% (131.88% proof) 1-20^16+20^(2*16)-20^(3*16)+20^(4*16) is prime! (0.0102s+0.0006s) a(8)=Phi[10,26^20]=157713125193403080417654809073419921485591\ 4988749414250433529072806660344375821341361193668309978857119\ 00082176001 OpenPFGW proof:$ ./pfgw -tc GGF_n5_20
PFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14]

Primality testing 1-26^20+26^(2*20)-26^(3*20)+26^(4*20) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N-1 test using base 23
Running N+1 test using discriminant 43, base 3+sqrt(43)
Calling N-1 BLS with factored part 47.34% and helper 1.33% (143.88% proof)
1-26^20+26^(2*20)-26^(3*20)+26^(4*20) is prime! (0.0178s+0.0004s)

a(9)=Phi[10,25^25]=6223015277861141707144064053780124170525328\
95248031358691463745029389305888319156836996164700086470525363\
32466843305155634880065917968750001

OpenPFGW proof:

$./pfgw -tc GGF_n5_25 PFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14] Primality testing 1-25^25+25^(2*25)-25^(3*25)+25^(4*25) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 13 Running N+1 test using discriminant 19, base 6+sqrt(19) Calling N-1 BLS with factored part 43.53% and helper 0.22% (131.03% proof) 1-25^25+25^(2*25)-25^(3*25)+25^(4*25) is prime! (0.0191s+0.0004s) a(10)=Phi[10,10^32]=9999999999999999999999999999999900000000000\ 000000000000000000000999999999999999999999999999999990000000000\ 0000000000000000000001 OpenPFGW proof:$ ./pfgw -tc GGF_n5_32
PFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14]

Primality testing 1-10^32+10^(2*32)-10^(3*32)+10^(4*32) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 29
Running N-1 test using base 31
Running N-1 test using base 41
Running N+1 test using discriminant 53, base 13+sqrt(53)
Calling N-1 BLS with factored part 40.94% and helper 1.88% (124.71% proof)
1-10^32+10^(2*32)-10^(3*32)+10^(4*32) is prime! (0.0205s+0.0004s)

• leizhou says:

a(51)=Phi[10,40842^12800]=1-40842^12800+40842^(2*12800)-40842^(3*12800)+40842^(4*12800)

236,089 digits

Found Prime Factors of p-1:
2
3
5
11
17
31
37
41
47
61
71
79
101
109
151
193
257
353
401
601
641
769
1201
1321
1601
2269
3361
4481
4801
6481
12289
14081
14401
25601
33601
40841
40961
54401
74561
82721
114689
133121
206501
218657
307201
765041
1652801
1857701
2966017
8232001
10320641
17377301
22958081
23243489
38031361
101171201
141260401
511796801
580208641
592472281
767813201
4696459711
7041703969
9717337601
10501723201
30515641601
248766146561
354276008321
517894582651
705146584001
27918237849409
160801300019551
251196515293121
740504382126701
1128341342943233
1264146271265531
1924021172026601
10305432550837001
106840174981862401
282638771475032321
299497698368208401
7874946188317929601
133466665536812834753
32579176238039271196801
103265713013107051982851
1644387444162459121795073
46281551799869591412662641
319850984076845481511828001
14699143858589845426640521217
78540254584496316212839437217
1231634760396384628753151465580001
25383772588726059050728090710020861
183459513338611775699980114423258801
8129138624812868079591688688707879493026042201
9275682700520645403799349823969863930448789546895316498950301
11066883583648857438527369426571814622297079595137168593270663\
19376481
30216334046792383030527184376489973772011567890018885884802183\
24728363758132963627347421437299209974795673601
30896784238673486628299380131568108740517922099602228223937912\
1654281362124763213189531902075674942016908420192175326593
23851072408931790360188409526856266289770692324409614279309066\
00452874800748867479431258764171876036353371143813835794080513\
04523248222661872160422948553076600478109632569542854540840281\
01871806398997299827538988902231528807845227971251407222277435\
232843063645443429377889441
16480153972425028814652469686408236011634578739365641597063187\
37731079127643300040264245664850954427681387172479374422310847\
81685121572472620611564378433984982118389957132207049685766073\
49603868943574181572523135989787435577230127856660414326593267\
88455123968449069668625422859575663901329301797820552273432415\
03804021818477170067381547596461842334296004284369092597561465\
70809114831796266872900264330655918914984699132380404783136881\
70373995044614922811578736747639294733439039004819202125065732\
61345272089987259225603363226848675681000647878131813309627851\
41364106440758072760868244679349198872853058925706321703793881\
49550200968475007800667576439785778527388964249545717524323455\
58362082507995417409577001139153249725059759485682602890753540\
97533644047284268966090655542051382789188583397277621968364899\
54102809943859773018778332594796485699966330598153583924964970\
44818595850171331925392382029871328044806068790672437221233956\
17355276430273648202173773280157312549893126057860674517258437\
03117419358866964727796384484090696149155161631362474217958722\
37882685894806587187775696923599858376523507069249743264222061\
91607970511151588013379476560937088032354757854995950903341566\
93739939367806185170471583646529196526818375135986407740957190\
09228664749953790623885918250696091619914721167204213311642242\
18668669305532418281906168988872304597055500400791351454099706\
97951257395699521484133136968724540108843554440294238313929772\
33616049319893978206265220867499862506536683832214261411274267\
34486415904607724720693877232783027058898579828650433852496554\
261222648182611186304277023950356998047432338870211390841118008\
44331678704978430003333134443706087404064882298451266128888895\
85456410385880982625258186313796883913013423980899492536116880\
92419315614538563511812238711232337081143942876417091789093613\
57883969235290895833685924976621249375523108503237670194723512\
56285241148360541164220189404969617802128097547984385828142083\
31301469529577275427046262270324650645981574336321288408793832\
03302579456420592773741200285446142652306006670677946472101846\
382731491389689697144520222534762438727016761119314478363872287\
32667589999841454172402905072756232275360710029263424374748050\
77599962082061644258698933189011811858822150367449116415765429\
31715427782365972831049380049162177851062155644761873671146391\
98493528840036847567668596454560693928236339921394224847080489\
24506553347005524008801330108270572887191147007607381493066174\
04696013352739461047216781355847296380386750498719702236727928\
37272999812187375601997568428182817024616193467591156837171088\
57018317636743156947111349281824208794646037657643857576808959\
45907252934578164723220051685648873847269325170806226786977256\
15342547203237811086240660769025679495782016160518535665992481\
20320498035818365706402337022971228398679211759541355421908563\
43982984088825455189853056900405180571583367137308852559010734\
41736955283633574628089809563861743836251859255088977146000559\
62792637705944871865525012513443235578039233733501483114067892\
56267387909797238119289661841268823771825650800821409757366836\
71750484905276475862430236161551563242991808834036567710927234\
74601316938278617201986523173282067362707233841621734917618484\
58813564415759476742885910279104361329578512298152871132105650\
20157188838972392620233763990800791585012217456845879802214083\
65494855482095991004642139209009666179184208104003787386259271\
27986365676638498570623754180497371739457984605258187029576810\
54844776121281880562978773799274862304101643709263001559084803\
17648412713217448653178087099063398121965582405194843889580814\
96773568787143021013095614502500133473023182457715403181531079\
79492886761596524887615854840929460368426290092545118129567877\
64770161685355259927216998982736815827001357510009660699137834\
75147532046718228956513356411684122759795084990100199520907948\
21624500893139805807165194602330798232640158559769048135701825\
72882578397470723885923670870900737779222613112668790075054137\
67220252273257105789403241987413235413027523659260737070913904\
73951519218088491237203525792477650798936570475853811000025824\
75930183763275890140156265969059671114970898622233156025081124\
59301925851262288220493413901299241751078725933015767150576155\
58373891554052488264621633806838684057556938866084271404988839\
61848058950690322309760239966215355119928611580945248044148758\
99260963510031111378568076923949142464297985525801717978299674\
7424382657394465455917412424108271439471508895799291694770151\
3594103234620654889153124068812315605197685911724097709028624\
4233898388493539136351274137125080844715413837454208536730002\
2987074173564582019169675718876273533528147689708801464220476\
11549854580246923324691660134614827481611840325544085738643530\
44954331009718167440128727002809718118519663354412508208595130\
25568597901375516622956721829575514906243498726143669858517875\
83456276413871480743544710400723743498032145363672115758693108\
02932829981653424874080285576482620669237694230338338508697683\
28451291127249791800676543723561678898206983539889654100743938\
01155173408857921813789232187631067196655780163499824553274623\
44069665426368330310048146782613224090106649535023617422685411\
30368825756224308071876844312547528166646390100654341285048315\
85686837376934594513779649871242390114365616568356644110881961\
50659765192415271198275304229418832544556698247807213958641283\
00359495571360399099534182367985471227743010663573738750559281\
20826428459825379582351498320945093548636579165739265384460790\
83114372580217164510960807607827020377028261974231191615874174\
33945147221045134315037034759340982492509369923933454061939842\
01913298038871840701691458780564203887409256997633387041310324\
75979784325670533320537155193048198968775995147874176924935939\
64509180808476499992540287580729831707656218299214030508269098\
79699581722747669174457100744337319234914909450593190053105701\
61880316487752282246705641762780670012825010559449684347027546\
77233549405171877603023664414778672469119107690702355179120470\
782729466675201

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-40842^12800+40842^(2*12800)-40842^(3*12800)+40842^(4*12800) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N+1 test using discriminant 43, base 18+sqrt(43)
Calling N-1 BLS with factored part 28.14% and helper 0.01% (84.43% proof)
1-40842^12800+40842^(2*12800)-40842^(3*12800)+40842^(4*12800) is Fermat and Lucas PRP! (21046.6553s+0.0461s)

CHG proof screen output:
Welcome to the CHG primality prover!
————————————

Input file is: GGF_n5_12800.in
Certificate file is: GGF_n5_12800.out
Found values of n, F and G.
Number to be tested has 236089 digits.
Modulus has 66447 digits.
Modulus is 28.14456642% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 8, u = 3. Right endpoint has 36751 digits.
Done! Time elapsed: 115841234ms.
Running CHG with h = 8, u = 3. Right endpoint has 34173 digits.
Done! Time elapsed: 101440703ms.
Running CHG with h = 7, u = 2. Right endpoint has 29469 digits.
Done! Time elapsed: 18687125ms.
Running CHG with h = 7, u = 2. Right endpoint has 23544 digits.
Done! Time elapsed: 24636469ms.
Running CHG with h = 5, u = 1. Right endpoint has 18604 digits.
Done! Time elapsed: 6587875ms.
Running CHG with h = 5, u = 1. Right endpoint has 4270 digits.
Done! Time elapsed: 27882797ms.
A certificate has been saved to the file: GGF_n5_12800.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “GGF_n5_12800.in”.

Pol[1, 1] with [h, u]=[5, 1] has ratio=1.190649948 E-36750 at X, ratio=1.9246747
01 E-23942 at Y, witness=2.
Pol[2, 1] with [h, u]=[4, 1] has ratio=1.880528215 E-19061 at X, ratio=5.4498439
02 E-14335 at Y, witness=13.
Pol[3, 1] with [h, u]=[6, 2] has ratio=0.2520200341 at X, ratio=1.717364487 E-98
80 at Y, witness=3.
Pol[4, 1] with [h, u]=[6, 2] has ratio=4.186665477 E-7396 at X, ratio=1.87890509
6 E-11850 at Y, witness=2.
Pol[5, 1] with [h, u]=[8, 3] has ratio=1.000000000 at X, ratio=1.169013535 E-141
11 at Y, witness=7.
Pol[6, 1] with [h, u]=[8, 3] has ratio=1.169013535 E-14111 at X, ratio=2.7813639
85 E-7736 at Y, witness=7.

Validated in 105 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(50)=Phi[10,38361^12500]=1-38361^12500+38361^(2*12500)-38361^(3*12500)+38361^(4*12500)

229,195 digits

Proven PRP by OpenPFGW using known prime factors of p^2-1 as helper:

to do…

• leizhou says:

a(49)=Phi[10,1172^10240]=1-1172^10240+1172^(2*10240)-1172^(3*10240)+1172^(4*10240)

125,704 digits

Proven PRP by OpenPFGW using known prime factors of p^2-1 as helper:
Primality testing 1-1172^10240+1172^(2*10240)-1172^(3*10240)+1172^(4*10240) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 37
Running N+1 test using discriminant 43, base 9+sqrt(43)
Calling N-1 BLS with factored part 25.46% and helper 0.00% (76.37% proof)
1-1172^10240+1172^(2*10240)-1172^(3*10240)+1172^(4*10240) is Fermat and Lucas PRP! (8108.2495s+0.0238s)

Primality not yet proven.

• leizhou says:

a(48)=Phi[10,26677^10000]=1-26677^10000+26677^(2*10000)-26677^(3*10000)+26677^(4*10000)

177,046 digits

Found Prime Factors of p-1:
2
3
5
7
11
13
17
19
37
41
61
101
103
151
193
251
401
641
751
1321
1601
3169
4001
4801
8017
13339
14051
16001
24251
27457
28001
28751
47041
57751
143401
150001
172321
398473
420001
980801
2100001
2252251
13829251
28756801
42675001
61942501
67412801
80311529
113045321
177738751
407273731
450638501
638684161
1081055809
3080250001
3863572751
8746617281
15292850201
28153765121
388422421001
471321167857
1198815576401
11055909197471
13118218360001
58605485928001
197994600538751
3830171583953681
5009874437561401
6248873850342881
7408918927437761
16233201565938901
243920250158790001
1309453815177708401
3160252067112463241
21781776134426538101
33941973176226200689
106703775942265559401
8558687804810167483751
29475847760042518849361
159020644450537919537951
57897429995388304968007167632801
86997948125940293022701126598401
9164334308739459558946100961100912151
3556784984769544259956788611192581828901
39091202832922131643360246870555131749226097001
26490306106400113794573804937626712260976630827563347998619037551
130443007588504865575767813891392166376431391200034485995456259051308947187625880001
823573688872421061764217274922290882389248174577256380507147923059180686452405431554601
72920971673469895781936728666792650354527392092004198349020057882445348541778967261678853107541249540501484010862806661181874497
18242404871736536430467328709948116304434287154916901224176376336705789334167464425433386118667371004254730741798251730031264557857190267555787746297518776353010447556844102042342476143972688718595611468194906235441013188119983853957130790866497224768893725003654314766921889952869767319665506565564365818109994167563869840371312408790287101453234862290918921851235013528253624267611494491314022038096412354841511170019237469031748089751
95426259021137352847081958896976809662774053943528198266780367750924838072735655534666686178277404214027202620742129188166736200184051070249231593693894506787111728330057184296105319876519229830478453769895985772239617314823193218871317547635260365362956607210469810883761613662048287888673147172941482259441639320551714436593103611708850452417980584300609794453000153367817634944490427843242085699539010821682413898431635340140458088161827149258892492556464308709809424710786873392804122387257214476884929905471438483584635290526004355129267972616936551417495873308518822685919351383366683638137530437460482154282064253262156009101057085739593381227398871984923412507877040736493530741375511328767808889057052945409254907874546724218577515786111185324407730377884260587192462120039391676184903970450643765487071317522470269788823764356599942797232756797975379542848961549722335612973296108996076377075492929341710529864864330346770614103220754639743115149979558278729457629144606348775228936549332783663149776329339524771475621214755087672621431002102424347130012125999577351839555081854520104493083260458700507089864992707311897946267455986668815664581142516482854124686610257542084749738207539519154741534168143681429991331550853560062293543093311151794080127825987069549848601848351751522272289243985816245437108417430579824055909102617216816934038784535045467158576994124156123715903859212575203288115167737786407677899780597262270168514218563889142200480962248533871775092287286150364147487485151457389765469019973468819981932392999461869430662382327400110760855354273813732505706616740281516644813321560710919234720503647409195396295266991479141637712049162292627512343575555635557795943868611064125554108865743711216287292310643334328516978229323407893786233216934236697618085969487123766131435420182290033467940697705595735169234364071777819393251073729203972449130637940441010812272531443719732102759823679353809621713662764818960856391202058816823259210601171428304829643708999474717270108869451224894425181432741681186089183254803291633912637065688770209221480352293200552896467833644626443778283895806717059087134932657683446300816886216904524275341383062603208327150556405584306225078751

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-26677^10000+26677^(2*10000)-26677^(3*10000)+26677^(4*10000) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N+1 test using discriminant 43, base 16+sqrt(43)
Calling N-1 BLS with factored part 27.03% and helper 0.00% (81.10% proof)
1-26677^10000+26677^(2*10000)-26677^(3*10000)+26677^(4*10000) is Fermat and Lucas PRP! (8238.0153s+0.0274s)

CHG proof screen output:
(13:09) gp > \r CHG.GP
realprecision = 68509 significant digits (68500 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: GGF_n5_10000.in
Certificate file is: GGF_n5_10000.out
Found values of n, F and G.
Number to be tested has 177046 digits.
Modulus has 48083 digits.
Modulus is 27.15807366% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 12, u = 5. Right endpoint has 32800 digits.
Done! Time elapsed: 887778203ms.
Running CHG with h = 12, u = 5. Right endpoint has 31898 digits.
Done! Time elapsed: 277822531ms.
Running CHG with h = 12, u = 5. Right endpoint has 30893 digits.
Done! Time elapsed: 390246875ms.
Running CHG with h = 11, u = 4. Right endpoint has 29474 digits.
Done! Time elapsed: 35751047ms.
Running CHG with h = 11, u = 4. Right endpoint has 28143 digits.
Done! Time elapsed: 88164281ms.
Running CHG with h = 11, u = 4. Right endpoint has 26006 digits.
Done! Time elapsed: 144496141ms.
Running CHG with h = 9, u = 3. Right endpoint has 24242 digits.
Done! Time elapsed: 26380234ms.
Running CHG with h = 9, u = 3. Right endpoint has 22280 digits.
Done! Time elapsed: 30178016ms.
Running CHG with h = 7, u = 2. Right endpoint has 19469 digits.
Done! Time elapsed: 9736250ms.
Running CHG with h = 7, u = 2. Right endpoint has 16956 digits.
Done! Time elapsed: 9378187ms.
Running CHG with h = 5, u = 1. Right endpoint has 10635 digits.
Done! Time elapsed: 1836360ms.
A certificate has been saved to the file: GGF_n5_10000.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “GGF_n5_10000.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=1.279211285 E-7434 at X, ratio=6.37452171
8 E-18069 at Y, witness=2.
Pol[2, 1] with [h, u]=[7, 2] has ratio=2.687721337 E-36137 at X, ratio=1.1097587
21 E-12643 at Y, witness=2.
Pol[3, 1] with [h, u]=[6, 2] has ratio=0.750076477 at X, ratio=3.258971343 E-502
6 at Y, witness=2.
Pol[4, 1] with [h, u]=[9, 3] has ratio=3.102349260 E-2812 at X, ratio=3.99328959
4 E-8433 at Y, witness=2.
Pol[5, 1] with [h, u]=[7, 3] has ratio=3.425175749 E-5886 at X, ratio=4.03698168
7 E-5886 at Y, witness=3.
Pol[6, 1] with [h, u]=[11, 4] has ratio=5.77601828 E-1766 at X, ratio=8.09929460
E-7060 at Y, witness=3.
Pol[7, 1] with [h, u]=[9, 4] has ratio=6.27781974 E-4571 at X, ratio=6.36568827
E-8546 at Y, witness=2.
Pol[8, 1] with [h, u]=[10, 4] has ratio=1.647597169 E-394 at X, ratio=3.48202939
1 E-5327 at Y, witness=2.
Pol[9, 1] with [h, u]=[11, 5] has ratio=3.893160262 E-4008 at X, ratio=1.2791869
34 E-7093 at Y, witness=641.
Pol[10, 1] with [h, u]=[12, 5] has ratio=0.2872425837 at X, ratio=1.163599687 E-
5025 at Y, witness=2.
Pol[11, 1] with [h, u]=[12, 5] has ratio=7.558106306 E-4470 at X, ratio=1.222299
509 E-4510 at Y, witness=2.

Validated in 361 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(47)=Phi[10,2468^8192]=1-2468^8192+2468^(2*8192)-2468^(3*8192)+2468^(4*8192)

111,161 digits

Proven PRP by OpenPFGW with know factor of p^2-1 as helper:
Primality testing 1-2468^8192+2468^(2*8192)-2468^(3*8192)+2468^(4*8192) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 23
Running N+1 test using discriminant 41, base 8+sqrt(41)
Calling N-1 BLS with factored part 25.28% and helper 0.00% (75.85% proof)
1-2468^8192+2468^(2*8192)-2468^(3*8192)+2468^(4*8192) is Fermat and Lucas PRP! (5616.6347s+0.0419s)

Primality not yet proven.

• leizhou says:

a(46)=Phi[10,16647^8000]=1-16647^8000+16647^(2*8000)-16647^(3*8000)+16647^(4*8000)

135,083 digits

Proven PRP by OpenPFGW with helpers:
Primality testing 1-16647^8000+16647^(2*8000)-16647^(3*8000)+16647^(4*8000) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 47
Running N+1 test using discriminant 71, base 17+sqrt(71)
Calling N-1 BLS with factored part 26.07% and helper 0.00% (78.22% proof)
1-16647^8000+16647^(2*8000)-16647^(3*8000)+16647^(4*8000) is Fermat and Lucas PRP! (6580.6021s+0.0264s)

Primality not yet proven.

• leizhou says:

a(45)=Phi[10, 820^6400]=1-820^6400+820^(2*6400)-820^(3*6400)+820^(4*6400)

74,594 digits

Found Prime Factors of p-1:
2
3
5
7
11
13
17
37
41
61
101
151
257
337
401
601
641
821
1069
1601
3361
15361
15809
16001
23321
25601
26881
29501
82721
100801
102913
180161
392321
414721
499801
626929
721169
1948801
3808001
9484289
22125953
61180201
134684261
184983521
224001601
333390361
377737361
386863361
3602726657
8817024001
14618690561
26291173121
722392582081
1284677826401
7493068673537
22536863364481
32917514723329
184412908168961
247071198294241
421571099651521
1780100757469501
3248602105204801
30779144840480401
22715627361398955233
24498014449561717121
606569987731446884273
20358544447009903031201
262570740857366277991937
763133411624100184476001
7109416245345074070417409
1839487760361537497420932897
3079769700199052908576503041
22467428295816925407695464001
1173062199615229775008689787201
24755777994211248866267135034724001
2903093378215761168724423180124996353
6167043619637130020518576835511460561
433351328245229936694897984261284070001
581794941628026913131544425397821183041
560790740733517033068941948246354481927521
1053678581461615208342914505238982232458202689
41785118500134605640208000121658089045478240001
120409166824916039667391457468300471474026082881
125112326610140081456922627444686499147406570963577197351
167805822709984795496479702213039765335422550606766727115344991335393285915257151196992960501
11838439428921560591693782075513626343906542183197717944200623973279039007340021566023618430541921
703305654288906436958219379497467053005976572248120053216059964483063461725714964703578611672186155248028561616187521
978580293892699148341589725148778359159362004313215238197209350283027890266893995808145380107785260350506149448882636419913257972671954046919994917726778468731569547550713829740183709601

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-820^6400+820^(2*6400)-820^(3*6400)+820^(4*6400) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N-1 test using base 29
Running N+1 test using discriminant 43, base 16+sqrt(43)
Calling N-1 BLS with factored part 27.09% and helper 0.00% (81.26% proof)
1-820^6400+820^(2*6400)-820^(3*6400)+820^(4*6400) is Fermat and Lucas PRP! (2754.9721s+0.0121s)

CHG proof screen output:
(22:23) gp > \r CHG.GP
realprecision = 35006 significant digits (35000 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: GGF_n5_6400.in
Certificate file is: GGF_n5_6400.out
Found values of n, F and G.
Number to be tested has 74594 digits.
Modulus has 20148 digits.
Modulus is 27.01006789% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 13, u = 5. Right endpoint has 14151 digits.
Done! Time elapsed: 64073125ms.
Running CHG with h = 13, u = 5. Right endpoint has 13921 digits.
Done! Time elapsed: 39040078ms.
Running CHG with h = 13, u = 5. Right endpoint has 13554 digits.
Done! Time elapsed: 21896766ms.
Running CHG with h = 12, u = 5. Right endpoint has 13157 digits.
Done! Time elapsed: 41997266ms.
Running CHG with h = 12, u = 5. Right endpoint has 12604 digits.
Done! Time elapsed: 18521953ms.
Running CHG with h = 11, u = 4. Right endpoint has 12207 digits.
Done! Time elapsed: 8035125ms.
Running CHG with h = 11, u = 4. Right endpoint has 11564 digits.
Done! Time elapsed: 14261047ms.
Running CHG with h = 11, u = 4. Right endpoint has 10727 digits.
Done! Time elapsed: 19014546ms.
Running CHG with h = 9, u = 3. Right endpoint has 9835 digits.
Done! Time elapsed: 4601875ms.
Running CHG with h = 9, u = 3. Right endpoint has 8736 digits.
Done! Time elapsed: 14953250ms.
Running CHG with h = 7, u = 2. Right endpoint has 7146 digits.
Done! Time elapsed: 2208766ms.
Running CHG with h = 5, u = 1. Right endpoint has 4332 digits.
Done! Time elapsed: 270375ms.
A certificate has been saved to the file: GGF_n5_6400.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “GGF_n5_6400.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=2.555099086457431501 E-2954 at X, ratio=8.184178629924674230 E-7286 at Y, witness=19.
Pol[2, 1] with [h, u]=[7, 2] has ratio=4.430362152376191681 E-14571 at X, ratio=4.980846811764702776 E-5628 at Y, witness=19.
Pol[3, 1] with [h, u]=[9, 3] has ratio=1.9968289704275057198 E-7285 at X, ratio=1.6221110537527721809 E-4771 at Y, witness=2.
Pol[4, 1] with [h, u]=[7, 3] has ratio=1.7686188003150724504 E-1821 at X, ratio=9.779052622597695405 E-3297 at Y, witness=3.
Pol[5, 1] with [h, u]=[11, 4] has ratio=2.552962520451149853 E-3572 at X, ratio=4.027671464435174202 E-3572 at Y, witness=19.
new witness: 23
Pol[6, 1] with [h, u]=[9, 4] has ratio=7.116054474805947673 E-837 at X, ratio=2.5642304680180752406 E-3345 at Y, witness=23.
new witness: 23
Pol[7, 1] with [h, u]=[9, 4] has ratio=2.5642304680180752406 E-3345 at X, ratio=1.9341137019436646588 E-2575 at Y, witness=23.
Pol[8, 1] with [h, u]=[11, 5] has ratio=6.228407959812901578 E-399 at X, ratio=1.0499743801245253174 E-1984 at Y, witness=2.
new witness: 29
Pol[9, 1] with [h, u]=[11, 5] has ratio=3.209438404884783388 E-1921 at X, ratio=3.2990809692383145096 E-2767 at Y, witness=29.
Pol[10, 1] with [h, u]=[12, 5] has ratio=0.4105098426946462310 at X, ratio=1.2883844305385905222 E-1984 at Y, witness=29.
Pol[11, 1] with [h, u]=[12, 5] has ratio=4.582050656828750418 E-1789 at X, ratio=1.7715191973547210819 E-1837 at Y, witness=2.
Pol[12, 1] with [h, u]=[12, 5] has ratio=6.119090286936575517 E-1149 at X, ratio=1.1571768894282322366 E-1148 at Y, witness=5.

Validated in 94 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(44)=Phi[10,85835^6250]=1-85835^6250+85835^(2*6250)-85835^(3*6250)+85835^(4*6250)

123,342 digits

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-85835^6250+85835^(2*6250)-85835^(3*6250)+85835^(4*6250) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 29
Running N-1 test using base 37
Running N-1 test using base 79
Running N+1 test using discriminant 103, base 4+sqrt(103)
Calling N-1 BLS with factored part 25.82% and helper 0.01% (77.47% proof)
1-85835^6250+85835^(2*6250)-85835^(3*6250)+85835^(4*6250) is Fermat and Lucas PRP! (9298.9293s+0.0287s)

Primality not yet proven.

• leizhou says:

a(43)=Phi[10,18781^5120]=1-18781^5120+18781^(2*5120)-18781^(3*5120)+18781^(4*5120)

87,526 digits

Found Prime Factors of p-1:
2
3
5
7
11
17
41
109
193
241
257
313
449
521
641
2129
2683
3169
4481
4801
4993
9281
9391
15329
16001
62081
87041
95177
115201
1123841
1575991
2755141
46491281
48906241
57960121
449302081
2146458241
135954940577
1351617784961
1519919506241
5543832524801
18977135105281
133887162150593
760271707658753
891331496412673
8389466967771649
25857830861383681
42087586476067432961
67676330893699091329
41786654405449262434561
13329331359075882949728001
53485558780722726795678433
17237463708283432044861971229329
15479242366153908770415703777906321
49000149510400908073294319016844801
156698418518470859845117003045437009333421256792221625729
1419862814692299906664212897075990686390699178466398669302977
239606945588735188192230280771961513254261340019541160410044802292641
25667702129305848151373544261920852104856347673532127443385632205510584072275605720273018108560631908722345351982231192806944345361
1466434752495962539949097289671993445321192633166678455856507074359430291443317014714663982720699780612055639271603389931736997622917542961259878859440077561150124826979528557265847157711069658504501648854019922286143274875259412788735261391624270316995563432412091521

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-18781^5120+18781^(2*5120)-18781^(3*5120)+18781^(4*5120) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 23
Running N+1 test using discriminant 37, base 12+sqrt(37)
Calling N-1 BLS with factored part 26.24% and helper 0.01% (78.74% proof)
1-18781^5120+18781^(2*5120)-18781^(3*5120)+18781^(4*5120) is Fermat and Lucas PRP! (2788.4635s+0.0535s)

Primality not yet proven.

• leizhou says:

a(42)=Phi[10,2779^5000]=1-2779^5000+2779^(2*5000)-2779^(3*5000)+2779^(4*5000)

68,878 digits

Found Prime Factors of p-1:
2
3
5
7
11
17
41
53
61
97
101
139
251
397
401
463
1409
1777
1801
2251
3001
9781
26251
28001
45737
52361
70001
77201
85201
165001
177601
280001
314651
920833
2110001
2200301
3421751
6625001
7204501
10207501
17868001
21135901
21561251
22845401
30548641
38353729
40101251
53954321
99999401
206726801
502168801
852246001
1032553321
1397032801
1084014891931
2385847627001
3537417052501
7396176017501
13972379830001
20154157730501
24277376884001
32270407337801
109157220707501
226249074206401
6532974698346881
7732923299076251
15672950674328501
45136319044665001
51749431494525001
1112436150131855561
1297513681431824951
10685011103812300001
22086697520381696201
742822421896253565438881251
3131388976925284151934442001
1787268072266115746172652964501
9265862453343489881881505918369
865939864818410533646043938554547101
17802667340444960623059295991533399466226401
244730009322447649521398891744226482312057735396101
150938813307175499555267788970607494149709109397792650739043068520201
4444312623136610776993589942272930472802123412041107017169910168232564704084222758166856885872117218109018905473215796704111096699450665549751890711035080958816222888340864444036018174419939902964022800266239907167788536015352567264577235136323786571047650640569012260215329240048682941830187684528150916001

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-2779^5000+2779^(2*5000)-2779^(3*5000)+2779^(4*5000) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 23
Running N+1 test using discriminant 79, base 20+sqrt(79)
Calling N-1 BLS with factored part 26.69% and helper 0.02% (80.11% proof)
1-2779^5000+2779^(2*5000)-2779^(3*5000)+2779^(4*5000) is Fermat and Lucas PRP! (2003.5697s+0.0119s)

CHG proof screen output:
(23:32) gp > \r CHG.GP
realprecision = 28003 significant digits (28000 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: GGF_n5_5000.in
Certificate file is: GGF_n5_5000.out
Found values of n, F and G.
Number to be tested has 68878 digits.
Modulus has 18387 digits.
Modulus is 26.69395296% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 16, u = 7. Right endpoint has 13720 digits.
Done! Time elapsed: 265814813ms.
Running CHG with h = 15, u = 6. Right endpoint has 13438 digits.
Done! Time elapsed: 224151547ms.
Running CHG with h = 15, u = 6. Right endpoint has 13076 digits.
Done! Time elapsed: 185175843ms.
Running CHG with h = 15, u = 6. Right endpoint has 12822 digits.
Done! Time elapsed: 148578610ms.
Running CHG with h = 15, u = 6. Right endpoint has 12545 digits.
Done! Time elapsed: 132095047ms.
Running CHG with h = 15, u = 6. Right endpoint has 12291 digits.
Done! Time elapsed: 87319093ms.
Running CHG with h = 13, u = 5. Right endpoint has 11844 digits.
Done! Time elapsed: 32560032ms.
Running CHG with h = 13, u = 5. Right endpoint has 11539 digits.
Done! Time elapsed: 31173968ms.
Running CHG with h = 13, u = 5. Right endpoint has 11154 digits.
Done! Time elapsed: 45009844ms.
Running CHG with h = 11, u = 4. Right endpoint has 10461 digits.
Done! Time elapsed: 35418641ms.
Running CHG with h = 11, u = 4. Right endpoint has 9943 digits.
Done! Time elapsed: 12198203ms.
Running CHG with h = 9, u = 3. Right endpoint has 9103 digits.
Done! Time elapsed: 6762484ms.
Running CHG with h = 9, u = 3. Right endpoint has 7997 digits.
Done! Time elapsed: 15702578ms.
Running CHG with h = 7, u = 2. Right endpoint has 6049 digits.
Done! Time elapsed: 1419907ms.
Running CHG with h = 5, u = 1. Right endpoint has 2175 digits.
Done! Time elapsed: 218843ms.
A certificate has been saved to the file: GGF_n5_5000.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “GGF_n5_5000.in”.

Pol[1, 1] with [h, u]=[5, 1] has ratio=1.506194722 E-13719 at X, ratio=5.3727582
76 E-7196 at Y, witness=2.
Pol[2, 1] with [h, u]=[7, 2] has ratio=1.813215239 E-8854 at X, ratio=4.71293358
8 E-7749 at Y, witness=2.
Pol[3, 1] with [h, u]=[9, 3] has ratio=4.039670426 E-4427 at X, ratio=1.78561613
6 E-5843 at Y, witness=2.
Pol[4, 1] with [h, u]=[7, 3] has ratio=6.67014851 E-1107 at X, ratio=2.967607850
E-3319 at Y, witness=2.
Pol[5, 1] with [h, u]=[11, 4] has ratio=4.007572251 E-3363 at X, ratio=6.4433914
4 E-3363 at Y, witness=3.
Pol[6, 1] with [h, u]=[11, 4] has ratio=1.291416094 E-519 at X, ratio=1.02802902
9 E-2071 at Y, witness=7.
Pol[7, 1] with [h, u]=[10, 5] has ratio=1.271819071 E-1529 at X, ratio=7.0691727
3 E-3464 at Y, witness=2.
Pol[8, 1] with [h, u]=[10, 5] has ratio=7.06917273 E-3464 at X, ratio=1.06518979
4 E-1924 at Y, witness=2.
Pol[9, 1] with [h, u]=[11, 5] has ratio=1.038528717 E-306 at X, ratio=3.83454608
0 E-1529 at Y, witness=2.
Pol[10, 1] with [h, u]=[12, 6] has ratio=2.306583942 E-1470 at X, ratio=4.267611
188 E-2679 at Y, witness=97.
Pol[11, 1] with [h, u]=[13, 6] has ratio=7.10033918 E-256 at X, ratio=4.24095980
9 E-1529 at Y, witness=5.
Pol[12, 1] with [h, u]=[13, 6] has ratio=2.092088597 E-277 at X, ratio=7.0879227
1 E-1658 at Y, witness=7.
Pol[13, 1] with [h, u]=[14, 6] has ratio=0.886434828 at X, ratio=1.859092352 E-1
529 at Y, witness=2.
Pol[14, 1] with [h, u]=[14, 6] has ratio=2.892455884 E-445 at X, ratio=1.8153420
94 E-2171 at Y, witness=13.
Pol[15, 1] with [h, u]=[16, 7] has ratio=1.000000000 at X, ratio=6.96875284 E-19
73 at Y, witness=2.

Validated in 163 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(41)=Phi[10,3928^4096]=1-3928^4096+3928^(2*4096)-3928^(3*4096)+3928^(4*4096)

58,887 digits

Found Prime Factors of p-1:
2
3
5
7
11
17
37
89
193
257
491
1217
3929
19993
40961
83401
114689
4086097
133788241
11793573889
14078503598209
111082404027713
23386790571808774657
370760798876308557313
13869575236597460275121
134108603374071427574273
8048468319322832665061383169
48689028769568233798223609857
2682236259321778369585486936736771364353
228132505214410879542555024857903200716687233
4149227443751427297626882942806154977325848502647086986889004979746532929
1893481730192957248452290402343976967887807844346104926503527348268833307249842563350237280537598946693964577266897248688930672134773536141642849266349596507552996212433606896531864912120051057598074484445503630501209903802900083469821078781125053705072212312076912322998255917152289057296427463479522247951370097074600581610170289635737762310274331505557139252376756405498105489804211449508378726902217688749771873594642953259554352753109475065475232628463805744400615471397737053081809299798489310292101007070222139658036691012388054731512406254870901507523354607995839545873463293338798192646556512945818001482792833016436282797516345681422122125366573383755925449899396145465562018817262510831503434182621704289523722286687769358227776538235017937540406874265220330851986647845558962830817117093212881308161

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-3928^4096+3928^(2*4096)-3928^(3*4096)+3928^(4*4096) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N+1 test using discriminant 29, base 10+sqrt(29)
Calling N-1 BLS with factored part 27.04% and helper 0.00% (81.11% proof)
1-3928^4096+3928^(2*4096)-3928^(3*4096)+3928^(4*4096) is Fermat and Lucas PRP! (1130.0777s+0.0079s)

CHG proof screen output:
(21:13) gp > \r CHG.GP
realprecision = 30006 significant digits (30000 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: GGF_n5_4096.in
Certificate file is: GGF_n5_4096.out
Found values of n, F and G.
Number to be tested has 58887 digits.
Modulus has 15921 digits.
Modulus is 27.03536904% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 12, u = 5. Right endpoint has 11127 digits.
Done! Time elapsed: 20162047ms.
Running CHG with h = 12, u = 5. Right endpoint has 10919 digits.
Done! Time elapsed: 19470734ms.
Running CHG with h = 12, u = 5. Right endpoint has 10554 digits.
Done! Time elapsed: 46600922ms.
Running CHG with h = 13, u = 5. Right endpoint has 10208 digits.
Done! Time elapsed: 33252140ms.
Running CHG with h = 11, u = 4. Right endpoint has 9678 digits.
Done! Time elapsed: 8884235ms.
Running CHG with h = 11, u = 4. Right endpoint has 9214 digits.
Done! Time elapsed: 6509859ms.
Running CHG with h = 11, u = 4. Right endpoint has 8468 digits.
Done! Time elapsed: 11340953ms.
Running CHG with h = 9, u = 3. Right endpoint has 7758 digits.
Done! Time elapsed: 2931063ms.
Running CHG with h = 9, u = 3. Right endpoint has 6900 digits.
Done! Time elapsed: 11030297ms.
Running CHG with h = 7, u = 2. Right endpoint has 5674 digits.
Done! Time elapsed: 1185984ms.
Running CHG with h = 5, u = 1. Right endpoint has 3503 digits.
Done! Time elapsed: 96375ms.
A certificate has been saved to the file: GGF_n5_4096.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “GGF_n5_4096.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=1.693820586 E-2392 at X, ratio=5.48637156
4 E-5895 at Y, witness=13.
Pol[2, 1] with [h, u]=[7, 2] has ratio=1.990945915 E-11789 at X, ratio=1.1471395
48 E-4342 at Y, witness=13.
Pol[3, 1] with [h, u]=[9, 3] has ratio=1.338600509 E-5894 at X, ratio=1.92652727
1 E-3677 at Y, witness=2.
Pol[4, 1] with [h, u]=[7, 3] has ratio=8.99936293 E-1474 at X, ratio=7.60208701
E-2576 at Y, witness=3.
Pol[5, 1] with [h, u]=[11, 4] has ratio=4.750442238 E-2841 at X, ratio=3.9033536
52 E-2839 at Y, witness=2.
new witness: 29
Pol[6, 1] with [h, u]=[9, 4] has ratio=5.081541220 E-747 at X, ratio=6.66778842
E-2986 at Y, witness=29.
new witness: 29
Pol[7, 1] with [h, u]=[9, 4] has ratio=6.66778842 E-2986 at X, ratio=4.254841313
E-1856 at Y, witness=29.
new witness: 13
Pol[8, 1] with [h, u]=[11, 5] has ratio=2.271398879 E-531 at X, ratio=6.04598459
E-2654 at Y, witness=13.
Pol[9, 1] with [h, u]=[12, 5] has ratio=0.999943782 at X, ratio=2.778060622 E-17
30 at Y, witness=2.
Pol[10, 1] with [h, u]=[12, 5] has ratio=1.659633395 E-793 at X, ratio=5.2783500
02 E-1825 at Y, witness=2.
Pol[11, 1] with [h, u]=[12, 5] has ratio=1.625164325 E-1148 at X, ratio=1.318848
601 E-1039 at Y, witness=2.

Validated in 53 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(40)=Phi[10,86^4000]=1-86^4000+86^(2*4000)-86^(3*4000)+86^(4*4000)

30,952 digits

Found Prime Factors of p-1:
2
3
5
11
13
17
29
41
43
101
151
191
251
281
401
569
641
1201
1249
1601
2081
3001
3581
4001
4261
7129
7673
13001
16001
17729
18121
25601
36901
57601
61057
77201
103001
149921
326881
699001
889001
1120001
1318831
1798001
3776161
5569001
7495361
37758001
91344401
20507296001
68201110001
265128253121
792970889251
18173596632001
399087586312601
1817313671510401
2991774758524141
3111323699168501
7104249490808801
8955690842006401
44046937552756951
490831741730341601
758831549911415401
1609456610239165801
5034100672099673501
5408926990383017561
15878663489053279751
226751243437978554901
1145789395950119269201
404323088012796038947297
38724695378340660516314413001
34496299175392289556651522734463439601
3366059601815649045436649795054866499701
5139085047092088517490765403718504985020066791359521
49709028710750003543435278852111166394952195556754204355176573946588823120641
5993377941415762392719244504162193722589771326885065538900335012896026265073857620555048799421411241725401
28173780346657203796913888277260380209208129923678982068145832991498709673623906239522153843584288533657894104709208550087008376739836546616513846783427111734883899333196770516598283103633408001
2659166242276148655755566188114115575190829924173430092081717423231983413839186868020869634690776773750987931947909507571918302413360025226943432114946397943277998760581107912215778599356388627466383671592976503002323987005380982361104017410563666036943082329059849156771823717984534874467908785291416071758888952919197725856257868347889887985617459006615955256155953224737271926878135139140544240737278697068925511648833740387899785004037226151622967526141225444700566060624292731130248314241

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-86^4000+86^(2*4000)-86^(3*4000)+86^(4*4000) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N-1 test using base 37
Running N+1 test using discriminant 47, base 1+sqrt(47)
Calling N-1 BLS with factored part 30.00% and helper 0.02% (90.02% proof)
1-86^4000+86^(2*4000)-86^(3*4000)+86^(4*4000) is Fermat and Lucas PRP! (462.7047s+0.0033s)

kp proof:
(13:16) gp >\r kppm.gp

(13:17) gp >N=1-86^4000+86^(2*4000)-86^(3*4000)+86^(4*4000)

(13:18) gp > lsm=[2^4000,3,5^4,11,13,17,29,41,43^4000,101,151,251,281,401,569,6
41,1201,1249,1601,2081,3001,3581,4001,7129,7673,13001,16001,17729,18121,25601,3
6901,57601,61057,77201,103001,149921,326881,699001,889001,1120001,1318831,17980
01,3776161,5569001,7495361,37758001,91344401,20507296001,68201110001,2651282531
21,792970889251,18173596632001,399087586312601,1817313671510401,299177475852414
1,3111323699168501,7104249490808801,8955690842006401,44046937552756951,49083174
1730341601,758831549911415401,1609456610239165801,5034100672099673501,540892699
0383017561,15878663489053279751,226751243437978554901,1145789395950119269201,40
4323088012796038947297,38724695378340660516314413001,34496299175392289556651522
734463439601,3366059601815649045436649795054866499701,5139085047092088517490765
403718504985020066791359521,497090287107500035434352788521111663949521955567542
04355176573946588823120641,5993377941415762392719244504162193722589771326885065
538900335012896026265073857620555048799421411241725401,281737803466572037969138
8827726038020920812992367898206814583299149870967362390623952215384358428853365
7894104709208550087008376739836546616513846783427111734883899333196770516598283
103633408001,265916624227614865575556618811411557519082992417343009208171742323
1983413839186868020869634690776773750987931947909507571918302413360025226943432
1149463979432779987605811079122157785993563886274663836715929765030023239870053
8098236110401741056366603694308232905984915677182371798453487446790878529141607
1758888952919197725856257868347889887985617459006615955256155953224737271926878
1351391405442407372786970689255116488337403878997850040372261516229675261412254
44700566060624292731130248314241]

(13:21) gp > kpm(lsm,N)

fraction = 300014/10^6
OK 0
OK 1
OK 2
OK 3
OK 4
OK 5

Round of root:
0
Root OK: above the round

Other roots are complex

Proof completed

• leizhou says:

a(39)=Phi[10,2257^3200]=1-2257^3200+2257^(2*3200)-2257^(3*3200)+2257^(4*3200)

42,926 dights

Found Prime Factors of p-1:
2
3
5
11
13
17
37
41
47
61
101
151
251
401
461
601
641
701
1129
1301
1601
1951
2753
3761
4001
8641
13217
20071
23041
25601
40801
53201
151201
430897
2376641
4174561
4727641
12147701
12194881
30110833
103415891
136248041
1293450131
1439182001
2599994401
5629772801
38673423361
68986911601
1595271285281
4746762270001
10639873343233
13821038811828251
694790762874403021
5693813735330468641
6487347886219314430201
6913277998972274633801
13258359614619024301233041
19254130352046048090062201
336683999034675896318894401
2157802276612593891611214547192961
5283418164478263844090018725382915901
71233898436943033520032054886611318301
884853443225812443563543686197799628466601
70080372442984922914358311945518873420164551
6230685755702062420755045810857408691512762401
637476564346566410795940897477752783843363407014185640606601
10774679908313519977545134022581161580809708462741399631015669638404596161
47639615524972464255551157325360460067167794583023079180597168621342554241
3161814255266921936208899722108082272957950925860459794765568393949218091958029434012887837862783272979579163347003826962267355206340958441727029541267634378775505830081
1986338805535688647921493079454894711775080111604262256577029450371702958139909758836525570864394496722167433422377978278409771446024168232704374811330645871019537521987800625555733407908527838756375297

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-2257^3200+2257^(2*3200)-2257^(3*3200)+2257^(4*3200) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 23
Running N+1 test using discriminant 53, base 13+sqrt(53)
Calling N-1 BLS with factored part 27.90% and helper 0.02% (83.72% proof)
1-2257^3200+2257^(2*3200)-2257^(3*3200)+2257^(4*3200) is Fermat and Lucas PRP! (746.1535s+0.0078s)

CHG proof screen output:
? \r CHG.GP
*** Warning: new stack size = 134217728 (128.000 Mbytes).
realprecision = 18013 significant digits (18000 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: GGF_n5_3200.in
Certificate file is: GGF_n5_3200.out
Found values of n, F and G.
Number to be tested has 42926 digits.
Modulus has 11976 digits.
Modulus is 27.897941151303339810% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 8, u = 3. Right endpoint has 7000 digits.
Done! Time elapsed: 7478207ms.
Running CHG with h = 8, u = 3. Right endpoint has 6849 digits.
Done! Time elapsed: 6977452ms.
Running CHG with h = 8, u = 3. Right endpoint has 6496 digits.
Done! Time elapsed: 8008796ms.
Running CHG with h = 8, u = 3. Right endpoint has 5673 digits.
Done! Time elapsed: 6583565ms.
Running CHG with h = 7, u = 2. Right endpoint has 5122 digits.
Done! Time elapsed: 2021607ms.
Running CHG with h = 6, u = 2. Right endpoint has 4002 digits.
Done! Time elapsed: 1048425ms.
Running CHG with h = 5, u = 1. Right endpoint has 3134 digits.
Done! Time elapsed: 117813ms.
A certificate has been saved to the file: GGF_n5_3200.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “GGF_n5_3200.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=3.009669380553270176 E-2479 at X, ratio=7.784452264251719249 E-3212 at Y, witness=2.
Pol[2, 1] with [h, u]=[6, 2] has ratio=0.9889822777969200736 at X, ratio=3.691272490470477510 E-1737 at Y, witness=3.
Pol[3, 1] with [h, u]=[6, 2] has ratio=0.6419154357707973634 at X, ratio=1.9642963334898130418 E-2239 at Y, witness=5.
Pol[4, 1] with [h, u]=[8, 3] has ratio=0.3391669381633492307 at X, ratio=2.644655266480345063 E-1656 at Y, witness=2.
Pol[5, 1] with [h, u]=[8, 3] has ratio=5.511660561571514141 E-523 at X, ratio=9.262707663718893510 E-2470 at Y, witness=2.
Pol[6, 1] with [h, u]=[8, 3] has ratio=9.262707663718893510 E-2470 at X, ratio=6.964463973308996652 E-1059 at Y, witness=2.
Pol[7, 1] with [h, u]=[8, 3] has ratio=6.964463973308996652 E-1059 at X, ratio=3.192234059047231211 E-454 at Y, witness=2.

Validated in 4 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(38)=Phi[10,32275^3125]=1-32275^3125+32275^(2*3125)-32275^(3*3125)+32275^(4*3125)

56,361 digits

Found Prime Factors of p-1:
2
3
5
11
31
41
101
163
251
281
401
421
509
1291
1601
5501
1023257
1040021
73983001
79945231
138060001
260070401
3254737001
11577146501
201180385501
4159526362501
6977596295251
13583237822501
220954647372774901
1717541566430982401
4894412966631502361401
7612558665685431857380001
173263826089030865775453751
295346926868617598782692901
102197396092630457140000170140881
16048515791932191725479923121848601
68762627120722833256055060752107193169615701
102442543159623421332789371521547570310955710955250394636239222529925552405647859601

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-32275^3125+32275^(2*3125)-32275^(3*3125)+32275^(4*3125) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 7
Running N-1 test using base 17
Running N+1 test using discriminant 23, base 1+sqrt(23)
Calling N-1 BLS with factored part 25.80% and helper 0.01% (77.41% proof)
1-32275^3125+32275^(2*3125)-32275^(3*3125)+32275^(4*3125) is Fermat and Lucas PRP! (1613.9501s+0.0112s)

Primality is not yet proven.

• leizhou says:

a(37)=Phi[10,14103^2560]=1-14103^2560+14103^(2*2560)-14103^(3*2560)+14103^(4*2560)

42,489 digits

Found Prime Factors of p-1:
2
3
5
11
17
31
37
41
43
193
241
257
281
401
641
769
881
1567
1913
2273
10433
11777
12161
13313
17011
17569
17921
62401
139121
251701
2749441
4432081
20436041
495301393
2135233537
3286304321
9171932411
11625092041
34732059521
271007938241
792323173633
2325664023211
22554567910634113
52482110640559073
106964754732090641
169121124570983681
818163588004343281
18741761*33132733441
179093575461637613013355084049
339931045080266765214688812648961
79829945009765837589787722877350261377
11587140952563526292080873297698373285219553
497244841793123959160325009688288357868843057106573083081
17275000133394653309301477856363233661998468890126271515846072137277424889090109499927715841

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-14103^2560+14103^(2*2560)-14103^(3*2560)+14103^(4*2560) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 23
Running N+1 test using discriminant 53, base 13+sqrt(53)
Calling N-1 BLS with factored part 26.40% and helper 0.02% (79.21% proof)
1-14103^2560+14103^(2*2560)-14103^(3*2560)+14103^(4*2560) is Fermat and Lucas PRP! (758.3227s+0.0074s)

CHG proven ended up with “Type” error:
? allocatemem(1024*1024*1024);
*** Warning: new stack size = 1073741824 (1024.000 Mbytes).
? \r chgcertd.gp
? CHGcertD(C)

Testing a PRP called “GGF_n5_2560.in”.

Pol[1, 1] with [h, u]=[7, 2] has ratio=1.9013502299010434628 E-2373 at X, ratio=3.297947752372408841 E-8263 at Y, witness=13.
Pol[2, 1] with [h, u]=[9, 3] has ratio=0.0004053358850669735711 at X, ratio=3.021162016990662409 E-5309 at Y, witness=2.
Pol[3, 1] with [h, u]=[11, 4] has ratio=0.06711560552140243117 at X, ratio=2.2558667297596616028 E-4248 at Y, witness=2.
Pol[4, 1] with [h, u]=[13, 5] has ratio=1.0000000000000000000 at X, ratio=2.3116069368088059594 E-3158 at Y, witness=2.
Pol[5, 1] with [h, u]=[13, 5] has ratio=2.3116069368088059594 E-3158 at X, ratio=5.828231687483148911 E-1699 at Y, witness=2.
Pol[6, 1] with [h, u]=[15, 6] has ratio=1.5785530250521719762 E-2765 at X, ratio=9.506574257439716351 E-2642 at Y, witness=7.
Pol[7, 1] with [h, u]=[15, 6] has ratio=9.506574257439716351 E-2642 at X, ratio=1.3596867843339354309 E-1132 at Y, witness=7.
“Type” error, so we quit!
Goodbye!

CHG certificate here.

• leizhou says:

a(36)=Phi[10,2692^2500]=1-3692^2500+3692^(2*2500)-3692^(3*2500)+3692^(4*2500)

35,673 digits

Found Prime Factors of p-1:
2
3
5
11
13
17
41
71
101
151
251
401
449
641
761
911
1151
1201
1231
3691
4253
8461
74101
96001
129841
539401
1027001
4900001
7420001
12360001
12406501
18283501
283518601
799232417
10597285001
13794000001
15656737651
15748687501
26244308801
42110664671
86590380001
153427548101
204007455731
584656151501
84046221037001
338152437795001
332871232800641
541347173900801
2169239338659751
2492277762264001
10485883747138001
26153709412818874961
31012460984886873281
30703847089792890395521
816246793844462803882665301
9068017859393131165435215701
17326509626426748165339029655647951
230256524210396641855458441480192251
175631211293225463144153534167775802001041
300855328008645425292372526836748232362001
77131448722595542394160116171984180244838801
1179682777877257335428704988179003639100097728068931505806201
337821334776883346726012378730653529733754906090095236991198388884331405558190776966767632463555201009904999412438686896850112656888954045554825053506565838243947772241687892160706714836442956161634556695990961917593355347576601597036126589771816501

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-3692^2500+3692^(2*2500)-3692^(3*2500)+3692^(4*2500) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 23
Running N-1 test using base 29
Running N-1 test using base 47
Running N-1 test using base 53
Running N+1 test using discriminant 73, base 2+sqrt(73)
Calling N-1 BLS with factored part 27.69% and helper 0.00% (83.09% proof)
1-3692^2500+3692^(2*2500)-3692^(3*2500)+3692^(4*2500) is Fermat and Lucas PRP! (822.3250s+0.0045s)

CHG proof screen output:
(12:22) gp > \r CHG.GP
realprecision = 15008 significant digits (15000 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: GGF_n5_2500.in
Certificate file is: GGF_n5_2500.out
Found values of n, F and G.
Number to be tested has 35673 digits.
Modulus has 9880 digits.
Modulus is 27.69489653% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 10, u = 4. Right endpoint has 6035 digits.
Done! Time elapsed: 4077672ms.
Running CHG with h = 9, u = 3. Right endpoint has 5664 digits.
Done! Time elapsed: 1910016ms.
Running CHG with h = 9, u = 3. Right endpoint has 5316 digits.
Done! Time elapsed: 2398719ms.
Running CHG with h = 8, u = 3. Right endpoint has 4721 digits.
Done! Time elapsed: 1836328ms.
Running CHG with h = 7, u = 2. Right endpoint has 4298 digits.
Done! Time elapsed: 611875ms.
Running CHG with h = 7, u = 2. Right endpoint has 3460 digits.
Done! Time elapsed: 466437ms.
Running CHG with h = 5, u = 1. Right endpoint has 2595 digits.
Done! Time elapsed: 110016ms.
A certificate has been saved to the file: GGF_n5_2500.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “GGF_n5_2500.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=1.642286443 E-1751 at X, ratio=5.31652420
8 E-2596 at Y, witness=17.
Pol[2, 1] with [h, u]=[7, 2] has ratio=1.869582438 E-5191 at X, ratio=5.71805339
E-1731 at Y, witness=17.
Pol[3, 1] with [h, u]=[6, 2] has ratio=0.871000565 at X, ratio=3.195837518 E-167
6 at Y, witness=7.
Pol[4, 1] with [h, u]=[7, 3] has ratio=1.079765428 E-1270 at X, ratio=1.19992152
3 E-1270 at Y, witness=5.
Pol[5, 1] with [h, u]=[8, 3] has ratio=1.000000000 at X, ratio=3.746204675 E-178
5 at Y, witness=7.
Pol[6, 1] with [h, u]=[8, 3] has ratio=3.746204675 E-1785 at X, ratio=2.27132388
3 E-1044 at Y, witness=7.
Pol[7, 1] with [h, u]=[10, 4] has ratio=1.271549249 E-535 at X, ratio=1.02983421
6 E-1483 at Y, witness=17.

Validated in 7 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(35)=Phi[10,13513^2048]=1-13513^2048+13513^(2*2048)-13513^(3*2048)+13513^(4*2048)

33,840 digits

Found Prime Factors of p-1:
2
3
5
17
29
97
193
233
257
277
563
881
2113
7297
9377
11777
13513
65921
209393
1038337
1087873
49495393
227703569
30275292161
211572744193
1571565256769
2174765564929
3472222834049
68567608774001
13737490965323777
69718647843932161
43359680849655741852737
112313944836428781593473793
56274345435895862569708282873972961281
3621133617552352168235443918242314447323716589441
6371280099442898545346674447215690383459056192732122014346477793

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-13513^2048+13513^(2*2048)-13513^(3*2048)+13513^(4*2048) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Running N+1 test using discriminant 19, base 19+sqrt(19)
Calling N-1 BLS with factored part 26.13% and helper 0.04% (78.44% proof)
1-13513^2048+13513^(2*2048)-13513^(3*2048)+13513^(4*2048) is Fermat and Lucas PRP! (374.9279s+0.0163s)

Primality not yet fully proven.

• leizhou says:

a(34)=Phi[10,7396^2000]=1-7396^2000+7396^(2*2000)-7396^(3*2000)+7396^(4*2000)

30,952 digits

a(34)=a(40)

• leizhou says:

a(33)=Phi[10,4965^1600]=1-4965^1600+4965^(2*1600)-4965^(3*1600)+4965^(4*1600)

23,654 digits

Found Prime Factors of p-1:
2
3
5
11
13
17
41
61
71
73
97
101
191
257
281
331
401
601
641
977
1201
1601
3361
4801
4751
5569
7681
19457
32411
40961
76801
90001
161377
605921
1164001
4008001
6439541
12325613
42481451
72739741
125997701
205420801
1429075001
1517827841
11660191901
19290677851
32352546353
1939613937601
8863368060241
12152275154689
29381250350761
55255028362631
60221179982609
303841447000313
227721414927737281
2838825266910381401
21986342394388058881
83224734749958760201
129775648607215982454611909643856351
4121009640214563638343288365513289761
4733145075050209953699119180107516655271986348101
125709442248195755241952051329039128676619631986529
8162344191991502431329669302097397642474525238118872675961205392737979767539802807655612246303815521
1669586317826076880171860486485852191033704922378091263900602751220565790451472814942927257304862399148564296728577
2506180249335441699127774960660089597261335411375841484550762222575294723783682176172029561738862339684796473499246597268468379913477360105200402131039557324527905905018025944627283379772719819893491511884801599201

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-4965^1600+4965^(2*1600)-4965^(3*1600)+4965^(4*1600) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N+1 test using discriminant 29, base 14+sqrt(29)
Calling N-1 BLS with factored part 29.10% and helper 0.00% (87.29% proof)
1-4965^1600+4965^(2*1600)-4965^(3*1600)+4965^(4*1600) is Fermat and Lucas PRP! (187.1466s+0.0037s)

CHG proof screen output:
? \r CHG.GP
*** Warning: new stack size = 134217728 (128.000 Mbytes).
realprecision = 10018 significant digits (10000 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1600/GGF_n5_1600.in
Certificate file is: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1600/GGF_n5_1600.out
Found values of n, F and G.
Number to be tested has 23654 digits.
Modulus has 6883 digits.
Modulus is 29.097131426894463570% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 6, u = 2. Right endpoint has 3007 digits.
Done! Time elapsed: 239662ms.
Running CHG with h = 6, u = 2. Right endpoint has 2559 digits.
Done! Time elapsed: 230620ms.
Running CHG with h = 5, u = 1. Right endpoint has 1802 digits.
Done! Time elapsed: 29845ms.
Running CHG with h = 5, u = 1. Right endpoint has 509 digits.
Done! Time elapsed: 21544ms.
A certificate has been saved to the file: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1600/GGF_n5_1600.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “/home/lzhou/prime/pfgw3.3.2/phis/CHG/1600/GGF_n5_1600.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=9.209816546041183153 E-1890 at X, ratio=7.196200953690116199 E-2398 at Y, witness=2.
Pol[2, 1] with [h, u]=[4, 1] has ratio=7.196200953690116199 E-2398 at X, ratio=1.0796018618774307017 E-1293 at Y, witness=2.
Pol[3, 1] with [h, u]=[6, 2] has ratio=0.8298617842906335697 at X, ratio=1.4874439603162354396 E-1516 at Y, witness=7.
Pol[4, 1] with [h, u]=[6, 2] has ratio=1.0714057919296877874 E-895 at X, ratio=1.6434501786990231190 E-895 at Y, witness=3.

Validated in 1 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(32)=Phi[10,1353^1280]=1-1353^1280+1353^(2*1280)-1353^(3*1280)+1353^(4*1280)

16,033 digits

Found Prime Factors of p-1:
2
3
5
11
13
17
41
61
241
257
353
641
677
941
1601
2281
3001
4481
7937
8761
10753
11071
37889
52121
321911
471041
833873
1416161
1474049
2233601
9489521
14269441
15941281
31426601
282799073
1321528421
1481236481
22142501441
28504257281
112067395201
1675564655441
3348654326161
1699556565744361
2186612675609249
396098815521624641
470022132123948961
41684087306595376001
748955699462058184001
1124027866654500468833
2720589381467633250749441
16823135947549123242427506544027029216503844686359479091201
2062450686415769415129666222365932122445402373284800695787217481921

Proven PRP by OpenPFGW using the above listed primes as helper:
Primality testing 1-1353^1280+1353^(2*1280)-1353^(3*1280)+1353^(4*1280) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N-1 test using base 43
Running N-1 test using base 47
Running N+1 test using discriminant 59, base 4+sqrt(59)
Calling N-1 BLS with factored part 28.05% and helper 0.07% (84.23% proof)
1-1353^1280+1353^(2*1280)-1353^(3*1280)+1353^(4*1280) is Fermat and Lucas PRP! (101.8846s+0.0021s)

CHG proof screen output:
? \r CHG.GP
*** Warning: new stack size = 134217728 (128.000 Mbytes).
realprecision = 8515 significant digits (8500 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1280/GGF_n5_1280.in
Certificate file is: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1280/GGF_n5_1280.out
Found values of n, F and G.
Number to be tested has 16033 digits.
Modulus has 4498 digits.
Modulus is 28.050803702899214990% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 8, u = 3. Right endpoint has 2542 digits.
Done! Time elapsed: 888600ms.
Running CHG with h = 8, u = 3. Right endpoint has 2417 digits.
Done! Time elapsed: 798060ms.
Running CHG with h = 8, u = 3. Right endpoint has 2127 digits.
Done! Time elapsed: 814510ms.
Running CHG with h = 7, u = 2. Right endpoint has 1912 digits.
Done! Time elapsed: 254677ms.
Running CHG with h = 6, u = 2. Right endpoint has 1474 digits.
Done! Time elapsed: 128431ms.
Running CHG with h = 5, u = 1. Right endpoint has 1138 digits.
Done! Time elapsed: 18437ms.
A certificate has been saved to the file: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1280/GGF_n5_1280.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “/home/lzhou/prime/pfgw3.3.2/phis/CHG/1280/GGF_n5_1280.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=3.1520651414572591526 E-959 at X, ratio=2.0696947399243466938 E-1225 at Y, witness=19.
Pol[2, 1] with [h, u]=[6, 2] has ratio=0.010869826920301646071 at X, ratio=1.6426500644927012385 E-673 at Y, witness=3.
Pol[3, 1] with [h, u]=[6, 2] has ratio=0.9260263764890115546 at X, ratio=1.3964025656569581056 E-876 at Y, witness=23.
Pol[4, 1] with [h, u]=[8, 3] has ratio=0.4607099170723954521 at X, ratio=3.996593309984639123 E-646 at Y, witness=5.
Pol[5, 1] with [h, u]=[8, 3] has ratio=2.382569974139252710 E-301 at X, ratio=9.242778133572669820 E-871 at Y, witness=7.
Pol[6, 1] with [h, u]=[8, 3] has ratio=9.242778133572669820 E-871 at X, ratio=1.3433868212851329615 E-373 at Y, witness=7.

Validated in 1 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here,

• leizhou says:

a(31)=Phi[10,2199^1250]=1-2199^1250+2199^(2*1250)-2199^(3*1250)+2199^(4*1250)

16,712 digits

Found Prime Factors of p-1:
2
3
5
7
11
31
41
61
101
157
241
251
281
601
701
733
1481
3251
9721
11701
17291
40351
201251
254291
230961301
16418972501
48512524961
3632295160001
134635401721501
196225588976251
453344624252501
592260051875651
5321857387463801
140674910350292501
190687752936590641
63071319844317116251
321180600673076967245251
38238162281492509256558153135940521
49426773666275497411922885281607398088921084263075446819451

Proven PRP by OpenPFGW:
Primality testing 1-2199^1250+2199^(2*1250)-2199^(3*1250)+2199^(4*1250) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N-1 test using base 19
Running N+1 test using discriminant 29, base 10+sqrt(29)
Calling N-1 BLS with factored part 27.09% and helper 0.02% (81.31% proof)
1-2199^1250+2199^(2*1250)-2199^(3*1250)+2199^(4*1250) is Fermat and Lucas PRP! (104.7165s+0.0025s)

CHG proof screen output:
? \r CHG.GP
*** Warning: new stack size = 536870912 (512.000 Mbytes).
realprecision = 7513 significant digits (7500 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1250/GGF_n5_1250.in
Certificate file is: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1250/GGF_n5_1250.out
Found values of n, F and G.
Number to be tested has 16712 digits.
Modulus has 4529 digits.
Modulus is 27.095818152800670032% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 12, u = 5. Right endpoint has 3128 digits.
Done! Time elapsed: 7193005ms.
Running CHG with h = 12, u = 5. Right endpoint has 3064 digits.
Done! Time elapsed: 6233990ms.
Running CHG with h = 12, u = 5. Right endpoint has 2967 digits.
Done! Time elapsed: 6557646ms.
Running CHG with h = 12, u = 5. Right endpoint has 2879 digits.
Done! Time elapsed: 6514438ms.
Running CHG with h = 11, u = 4. Right endpoint has 2750 digits.
Done! Time elapsed: 3431032ms.
Running CHG with h = 11, u = 4. Right endpoint has 2640 digits.
Done! Time elapsed: 3456465ms.
Running CHG with h = 11, u = 4. Right endpoint has 2431 digits.
Done! Time elapsed: 3524620ms.
Running CHG with h = 9, u = 3. Right endpoint has 2264 digits.
Done! Time elapsed: 956062ms.
Running CHG with h = 9, u = 3. Right endpoint has 2077 digits.
Done! Time elapsed: 952181ms.
Running CHG with h = 7, u = 2. Right endpoint has 1811 digits.
Done! Time elapsed: 188846ms.
Running CHG with h = 7, u = 2. Right endpoint has 1584 digits.
Done! Time elapsed: 176420ms.
Running CHG with h = 5, u = 1. Right endpoint has 1009 digits.
Done! Time elapsed: 11534ms.
A certificate has been saved to the file: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1250/GGF_n5_1250.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “/home/lzhou/prime/pfgw3.3.2/phis/CHG/1250/GGF_n5_1250.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=1.5164973054060442926 E-615 at X, ratio=4.782854108698667764 E-1624 at Y, witness=2.
Pol[2, 1] with [h, u]=[7, 2] has ratio=1.5130848967739664058 E-3247 at X, ratio=1.0253440778584114814 E-1150 at Y, witness=2.
Pol[3, 1] with [h, u]=[7, 2] has ratio=0.3023561729068097643 at X, ratio=2.783942218263344449 E-454 at Y, witness=2.
Pol[4, 1] with [h, u]=[9, 3] has ratio=9.899533456104065957 E-267 at X, ratio=3.720957293953139009 E-798 at Y, witness=3.
Pol[5, 1] with [h, u]=[7, 3] has ratio=1.5925358186093381206 E-564 at X, ratio=1.7938227384473093284 E-564 at Y, witness=5.
Pol[6, 1] with [h, u]=[11, 4] has ratio=4.439597470085346157 E-167 at X, ratio=1.3928188717244440722 E-665 at Y, witness=3.
Pol[7, 1] with [h, u]=[9, 4] has ratio=6.494744805202421687 E-251 at X, ratio=1.0432989025347971412 E-838 at Y, witness=2.
Pol[8, 1] with [h, u]=[10, 4] has ratio=0.8347627625351698111 at X, ratio=5.106842377319111524 E-439 at Y, witness=3.
Pol[9, 1] with [h, u]=[11, 5] has ratio=4.128433025505467712 E-130 at X, ratio=1.1992955375308464617 E-647 at Y, witness=41.
Pol[10, 1] with [h, u]=[12, 5] has ratio=0.5104843803898130217 at X, ratio=6.051206873773523831 E-439 at Y, witness=3.
Pol[11, 1] with [h, u]=[12, 5] has ratio=0.8926668496685095845 at X, ratio=5.732465572228929201 E-487 at Y, witness=3.
Pol[12, 1] with [h, u]=[12, 5] has ratio=4.522240678004094341 E-320 at X, ratio=7.854305408693594697 E-320 at Y, witness=3.

Validated in 3 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate here.

• leizhou says:

a(30)=Phi[10,951^1024]=1-951^1024+951^(2*1024)-951^(3*1024)+951^(4*1024)

12,199 digits

Found Prime Factors of p-1:
2
3
5
7
17
19
97
193
257
317
641
769
937
2729
34369
40961
159937
239873
452201
142245889
431628289
40559337473
2381472717313
5106434775041
769732632934162433
334513877809772987888801
15545435467778440779784750000980441948449
2914604628480219065024736459216074151868073183208103087472666364231463025588136369576766529

Proven PRP by OpenPFGW:
Primality testing 1-951^1024+951^(2*1024)-951^(3*1024)+951^(4*1024) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N-1 test using base 31
Running N+1 test using discriminant 41, base 8+sqrt(41)
Calling N-1 BLS with factored part 27.28% and helper 0.00% (81.83% proof)
1-951^1024+951^(2*1024)-951^(3*1024)+951^(4*1024) is Fermat and Lucas PRP! (34.0934s+0.0037s)

CHG proof screen output
? \r CHG.GP
*** Warning: new stack size = 134217728 (128.000 Mbytes).
realprecision = 8515 significant digits (8500 digits displayed)

Welcome to the CHG primality prover!
————————————

Input file is: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1024.2/GGF_n5_1024.in
Certificate file is: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1024.2/GGF_n5_1024.out
Found values of n, F and G.
Number to be tested has 12199 digits.
Modulus has 3328 digits.
Modulus is 27.276365863651275450% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 12, u = 5. Right endpoint has 2217 digits.
Done! Time elapsed: 5183013ms.
Running CHG with h = 11, u = 4. Right endpoint has 2138 digits.
Done! Time elapsed: 2443198ms.
Running CHG with h = 11, u = 4. Right endpoint has 2049 digits.
Done! Time elapsed: 2530304ms.
Running CHG with h = 10, u = 4. Right endpoint has 1951 digits.
Done! Time elapsed: 1607592ms.
Running CHG with h = 10, u = 4. Right endpoint has 1832 digits.
Done! Time elapsed: 1604879ms.
Running CHG with h = 9, u = 3. Right endpoint has 1715 digits.
Done! Time elapsed: 764758ms.
Running CHG with h = 9, u = 3. Right endpoint has 1597 digits.
Done! Time elapsed: 723254ms.
Running CHG with h = 9, u = 3. Right endpoint has 1431 digits.
Done! Time elapsed: 706179ms.
Running CHG with h = 7, u = 2. Right endpoint has 1204 digits.
Done! Time elapsed: 164830ms.
Running CHG with h = 5, u = 1. Right endpoint has 878 digits.
Done! Time elapsed: 40934ms.
A certificate has been saved to the file: /home/lzhou/prime/pfgw3.3.2/phis/CHG/1024.2/GGF_n5_1024.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “/home/lzhou/prime/pfgw3.3.2/phis/CHG/1024.2/GGF_n5_1024.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=3.297161725600899846 E-516 at X, ratio=7.305083966748162134 E-978 at Y, witness=2.
Pol[2, 1] with [h, u]=[7, 2] has ratio=3.529713476532110786 E-1955 at X, ratio=3.280296881135144090 E-652 at Y, witness=2.
Pol[3, 1] with [h, u]=[9, 3] has ratio=4.693217347032501564 E-228 at X, ratio=1.6448948476911944098 E-682 at Y, witness=7.
Pol[4, 1] with [h, u]=[7, 3] has ratio=3.701764132916840871 E-386 at X, ratio=1.6094814383703918343 E-497 at Y, witness=3.
Pol[5, 1] with [h, u]=[8, 3] has ratio=0.8734928290915691618 at X, ratio=2.200516331241117326 E-357 at Y, witness=23.
Pol[6, 1] with [h, u]=[9, 4] has ratio=5.716391202518547967 E-117 at X, ratio=1.0677947191781277243 E-465 at Y, witness=2.
Pol[7, 1] with [h, u]=[9, 4] has ratio=1.0677947191781277243 E-465 at X, ratio=3.027551804095253631 E-476 at Y, witness=2.
Pol[8, 1] with [h, u]=[10, 4] has ratio=1.0000000000000000000 at X, ratio=4.127131920631514080 E-393 at Y, witness=11.
Pol[9, 1] with [h, u]=[10, 4] has ratio=4.127131920631514080 E-393 at X, ratio=1.7458848636980266024 E-359 at Y, witness=11.
Pol[10, 1] with [h, u]=[12, 5] has ratio=0.6227501349948038056 at X, ratio=6.799002328964641506 E-396 at Y, witness=2.

Validated in 2 sec.

Congratulations! n is prime!
Goodbye!

CHG Certificate here.

• leizhou says:

a(29)=Phi[10,2866^1000]=1-2866^1000+2866^(2*1000)-2866^(3*1000)+2866^(4*1000)

13,830 digits

Found Prime Factors of p-1:
2
3
5
11
41
47
61
101
151
191
251
353
601
1433
1951
2251
3001
5501
23269
219281
671501
1858651
25057793
27961441
343890751
561625249
11313818501
88197048451
329232305291
5233946392151
67469073169937
4552075280222246897964373261
1472412635145433490530776216108429283681
73661292000510194783244623925134228900220266651
139041924655075182857231609496958379107237826803053151
20721394402220809402470170274305791519254668054259233521

Proven PRP by OpenPFGW:
Primality testing 1-2866^1000+2866^(2*1000)-2866^(3*1000)+2866^(4*1000) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N-1 test using base 23
Running N+1 test using discriminant 37, base 14+sqrt(37)
Calling N-1 BLS with factored part 27.77% and helper 0.08% (83.38% proof)
1-2866^1000+2866^(2*1000)-2866^(3*1000)+2866^(4*1000) is Fermat and Lucas PRP! (74.4629s+0.0020s)

CHG proof screen output:
(19:09) gp > \r examples\CHG

Welcome to the CHG primality prover!
————————————

Input file is: Y:\prime\pfgw3.3.2\phis\CHG\1000\GGF_n5_1000.in
Certificate file is: Y:\prime\pfgw3.3.2\phis\CHG\1000\GGF_n5_1000.out
Found values of n, F and G.
Number to be tested has 13830 digits.
Modulus has 3833 digits.
Modulus is 27.70999113% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 10, u = 4. Right endpoint has 2334 digits.
Done! Time elapsed: 1225297ms.
Running CHG with h = 9, u = 3. Right endpoint has 2182 digits.
Done! Time elapsed: 723328ms.
Running CHG with h = 9, u = 3. Right endpoint has 2026 digits.
Done! Time elapsed: 743203ms.
Running CHG with h = 8, u = 3. Right endpoint has 1824 digits.
Done! Time elapsed: 769969ms.
Running CHG with h = 7, u = 2. Right endpoint has 1655 digits.
Done! Time elapsed: 375484ms.
Running CHG with h = 7, u = 2. Right endpoint has 1331 digits.
Done! Time elapsed: 240047ms.
Running CHG with h = 5, u = 1. Right endpoint has 966 digits.
Done! Time elapsed: 41000ms.
A certificate has been saved to the file: Y:\prime\pfgw3.3.2\phis\CHG\1000\GGF_
n5_1000.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “Y:\prime\pfgw3.3.2\phis\CHG\1000\GGF_n5_1000.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=1.208991093 E-694 at X, ratio=4.52285866
E-1097 at Y, witness=13.
Pol[2, 1] with [h, u]=[7, 2] has ratio=1.353053792 E-2193 at X, ratio=1.10604217
5 E-731 at Y, witness=13.
Pol[3, 1] with [h, u]=[6, 2] has ratio=0.693099883 at X, ratio=4.614624124 E-649
at Y, witness=2.
Pol[4, 1] with [h, u]=[7, 3] has ratio=2.823157962 E-507 at X, ratio=3.219655275
E-507 at Y, witness=2.
Pol[5, 1] with [h, u]=[8, 3] has ratio=1.000000000 at X, ratio=6.51856104 E-607
at Y, witness=7.
Pol[6, 1] with [h, u]=[8, 3] has ratio=6.51856104 E-607 at X, ratio=3.131600253
E-467 at Y, witness=7.
Pol[7, 1] with [h, u]=[10, 4] has ratio=3.577591556 E-172 at X, ratio=1.42926488
9 E-608 at Y, witness=13.

Validated in 2 sec.

Congratulations! n is prime!
Goodbye!

CHG Certifiacte here.

• leizhou says:

a(28)=Phi[10,1274^800]=1-1274^800+1274^(2*800)-1274^(3*800)+1274^(4*800)

9,937 digits

Found Prime Factors of p-1:
2
3
5
7
11
13
17
19
41
67
101
137
401
701
1171
1409
2801
3191
3329
5153
9601
10601
11551
16001
25601
28351
30161
34721
88321
140891
1079681
1228001
1264129
1623077
7946581
8365061
44134241
108330721
469000481
946550321
1075362641
5799897553
6943615801
233772777473
160243615955201
8214188603744341
22858833064468801
6099204480446452801
1532418942996024103605679009
6936256826041956531609814245079225659001
791786009283161750926035868654189600872973040801
77487310953404247114192148492153103873846305261721101
1069432429185705895631586384273286243218040708194819227338622\
9980221185087771990245956180878180521774185615407294434619732\
4424055081944642500016244278316842015818413043429731801214348\
1711896290215617415629304537770497782279624601

PRP screen output from OpenPFGW:
Primality testing 1-1274^800+1274^(2*800)-1274^(3*800)+1274^(4*800) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 37
Running N-1 test using base 47
Running N-1 test using base 71
Running N+1 test using discriminant 79, base 15+sqrt(79)
Calling N-1 BLS with factored part 31.61% and helper 0.02% (94.86% proof)
1-1274^800+1274^(2*800)-1274^(3*800)+1274^(4*800) is Fermat and Lucas PRP! (45.0762s+0.0011s)

kp proof screen output:
\r kppm

N=1-1274^800+1274^(2*800)-1274^(3*800)+1274^(4*800)

lsm=[2^800,3,5^4,7^1600,11,13^800,17,19,41,67,101,137,401,701,1171,1409,2801,3191,33
29,5153,9601,10601,11551,16001,25601,28351,30161,34721,88321,140891,1079681,1228
001,1264129,1623077,7946581,8365061,44134241,108330721,469000481,946550321,10753
62641,5799897553,6943615801,233772777473,160243615955201,8214188603744341,228588
33064468801,6099204480446452801,1532418942996024103605679009,6936256826041956531
609814245079225659001,791786009283161750926035868654189600872973040801,774873109
53404247114192148492153103873846305261721101,10694324291857058956315863842732862
43218040708194819227338622998022118508777199024595618087818052177418561540729443
46197324424055081944642500016244278316842015818413043429731801214348171189629021
5617415629304537770497782279624601]

(11:17) gp > kpm(lsm,N)

fraction = 316140/10^6
OK 0
OK 1
OK 2
OK 3
OK 4
OK 5

Round of root:
0
Root OK: above the round

Other roots are complex

Proof completed

Certificates including the 229 digits factor of p-1 here.

• leizhou says:

a(27)=Phi[10,2123^640]=1-2123^640+2123^(2*640)-2123^(3*640)+2123^(4*640)

8,517 digits

Found Prime Factors of p-1:
2
3
5
11
17
31
41
59
193
257
449
641
821
1061
10993
77569
166273
547741
686321
1589377
1610681
3245569
3543041
97005121
1305284353
1330683337
1337920513
39853739777
89500053121
654988635251
20323784966761
183531851206432381
206333600987563689620107441
65232613565996134472134965102705869052257377
105728086185887951345406191545165949147162305001
52530108121785709366708672122186481758870221062184808081650884\
00288803288402041294588046251692267840330288692952077815757744\
11297174998284761241579999122872786344687506425909061980519837\
745337803173829569165121
16258151788407946298951839088634460456029119974632410105122826\
98503607312425542807349574260599935796558495086404957786102806\
34098957831235356350895119419946720924418082098389856838286123\
94317374369795973822528605364756093870139833649925726766207776\
82671320800652055319384618045919691487187453059719716266163610\
58971831596143303189481472804255245530297695007391421509323803\
94619717345914522793087925450760779011332714469181065168019167\
95726647323648199734547282766428474632433345099702742374507109\
91721239872852184198451079539476375978987044788060181965722690\
43486775012554169094248290334374797060167485456754835635503264\
78568928998934910044521426594560233273442561570008792669822930\
72058741385840034453945541384791704193981666295863185399520707\
51583612932942327382150002976369408870340726886748645342894984\
384691206208750647268353

Screen output of OpenPFGW proof:
Primality testing 1-2123^640+2123^(2*640)-2123^(3*640)+2123^(4*640) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N+1 test using discriminant 29, base 10+sqrt(29)
Calling N-1 BLS with factored part 40.68% and helper 0.06% (122.09% proof)
1-2123^640+2123^(2*640)-2123^(3*640)+2123^(4*640) is prime! (22.0369s+0.0011s)

Primo Certificates of the two large prime factors of p-1 are here.

• leizhou says:

a(26)=Phi[10,2336^625]=1-2336^625+2336^(2*625)-2336^(3*625)+2336^(4*625)

8,422 digits

Found Prime Factors of p-1:
2
5
11
73
401
467
3251
1086901
4193251
4444621
5456897
1479067501
541644849511
2587460310751
398496915729251
18314142035325601
199502294094588751621
29303360464845025432260814442119479626409918401951
3989076032726071335951142640182635992487960314909339838429698872\
1708185136216958629572397569230178977282805484154917662706224160\
5654238275984703891649931427182932594292026231625241204090170456\
0749374223567797001504038120082062349028981819224275818909429037\
5323106463028618000309109568343161918988693791794275638251

The 314 digits prime factor of p-1 is proven using PRIMO, certificate in the certificate pack.

PRP screen output from OpenPFGW:
Primality testing 1-2336^625+2336^(2*625)-2336^(3*625)+2336^(4*625) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 7
Running N+1 test using discriminant 17, base 8+sqrt(17)
Calling N-1 BLS with factored part 30.84% and helper 0.01% (92.53% proof)
1-2336^625+2336^(2*625)-2336^(3*625)+2336^(4*625) is Fermat and Lucas PRP! (24.1172s+0.0008s)

kp proof screen output:
\r kppm

N=1-2336^625+2336^(2*625)-2336^(3*625)+2336^(4*625)

lsm=[2^3125,5^5,11,73^625,401,467,3251,1086901,4193251,4444621,5456897,1479067501,541644849511,2587460310751,398496915729251,18314142035325601,199502294094588751621,2930336046484502543226
0814442119479626409918401951,3989076032726071335951142640182635992487960314909339838429698872170818513621695862957239756923017897728280548415491766270622416056542382759847038916499314
2718293259429202623162524120409017045607493742235677970015040381200820623490289818192242758189094290375323106463028618000309109568343161918988693791794275638251]

(19:48) gp > kpm(lsm,N)

fraction = 307820/10^6
OK 0
OK 1
OK 2
OK 3
OK 4
OK 5

Round of root:
-6278945804716475622674389778543577310952295855046343847469029637825566133759628
89694925881219900050141125619029837663846490434891919036290716570182426448812436
59338739824075585851285961326487267886289955435600568796129115780719429554746958
33279794982356437628417711401708047751867995315339054071603365028324385813718927
30785294603052513408462661284715171350973145347965067168669062810878226264449957
79213679854941319077960826651093848510932496493339894919788413591666214443720060
63544348131539466032539652763506545532514960681512374397062346128681588182600880
80821538580749401548934822853402423143947949322149279585801405582740159635476719
35422477251196452103034494412071629906452743520747280157174395936135494687236108
47394580115533283330682480815720316534391549155299778756038102499349518093101332
1637929509
Root OK: above the round

Round of root:
0
Root OK: below the round

Round of root:
62789458047164756226743897785435773109522958550463438474690296378255661337596288
96949258812199000501411256190298376638464904348919190362907165701824264488124365
93387398240755858512859613264872678862899554356005687961291157807194295547469583
32797949823564376284177114017080477518679953153390540716033650283243858137189273
07852946030525134084626612847151713509731453479650671686690628108782262644499577
92136798549413190779608266510938485109324964933398949197884135916662144437200606
35443481315394660325396527635065455325149606815123743970623461286815881826008808
08215385807494015489348228534024231439479493221492795858014055827401596354767193
54224772511964521030344944120716299064527435207472801571743959361354946872361084
73945801155332833306824808157203165343915491552997787560381024993495180931013321
637929509
Root OK: below the round

Proof completed

Certificate here.

• leizhou says:

a(25)=Phi[10,34^512]=1-34^512+34^(2*512)-34^(3*512)+34^(4*512)

3,137 digits

Equals to a(21)

• leizhou says:

a(24)=Phi[10,511^500]=1-511^500+511^(2*500)-511^(3*500)+511^(4*500)

5,417 digits

Found Prime Factors of p-1:
2
3
5
7
11
17
31
61
71
73
101
137
251
401
593
601
953
5521
7901
9001
9851
10501
22501
32401
306121
1138901
12282833
50383441
112922101
1420889501
58293427001
319142001889
734778061601
4005502932251
211742894078101
229631614132001
7725120944429201
126812256620356381
513452027872643501
2451039636224762310675352082651
1391999738492087545779388034427591879001
1679435125764571207199153088231147280805327515865168619978956630752707405601
13977084342571293872308109180963667991208015959390566489806990748380101845642395559448587001

Screen Output of OpenPFGW proof:

Primality testing 1-511^500+511^(2*500)-511^(3*500)+511^(4*500) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N-1 test using base 37
Running N+1 test using discriminant 43, base 18+sqrt(43)
Calling N-1 BLS with factored part 33.70% and helper 0.07% (101.17% proof)
1-511^500+511^(2*500)-511^(3*500)+511^(4*500) is prime! (9.7577s+0.0008s)

• leizhou says:

a(23)=Phi[10,647^400]=1-647^400+647^(2*400)-647^(3*400)+647^(4*400)

4,498 digits

Found Prime Factors of p-1:
2
3
5
11
17
19
31
41
61
73
101
193
241
401
647
701
1021
1801
3221
13697
14561
36353
420001
430193
511801
4512901
5359801
5453521
22590401
56724001
151779601
254316301
1200229417
3931644401
4407288001
5643970091
15954977651
171833389601
148089363924737
6903391930108601
17528421577675201
108020057546754701
504967281311738081
6912505576413737101
39737484959892211301
602313651062925384901
203328015737502170561281
34098250797899432902407124520298301
942906198444279251975404844847530188126023361
22750279788656768299977247892852668296972279521655752257
55291857184356732714502409132562488499965918078932045540961
16055247984496002837749572727600900255632071177599487981777013220401

OpenPFGW proof screen output:

Primality testing 1-647^400+647^(2*400)-647^(3*400)+647^(4*400) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N+1 test using discriminant 37, base 12+sqrt(37)
Calling N-1 BLS with factored part 38.43% and helper 0.15% (115.44% proof)
1-647^400+647^(2*400)-647^(3*400)+647^(4*400) is prime! (5.8490s+0.0007s)

• leizhou says:

a(22)=Phi[10,1619^320]=1-1619^320+1619^(2*320)-1619^(3*320)+1619^(4*320)

4,108 digits

Found Prime Factors of p-1:
2
3
5
11
17
41
61
71
97
233
251
281
809
1151
1619
2161
3433
9721
21569
23761
71329
77093
78593
704321
4294649
27764801
108462901
385765741
2501592001
4090999921
20973301121
81760188281
183683543920963649
3462504332700185321
300303869105695313297
3044057549445299254561
18873868836708273720221
876620181544868058248423393
1144016412612648573399025547136338219064064146226779787616067440801

OpenPFGW proof screen output:
Primality testing 1-1619^320+1619^(2*320)-1619^(3*320)+1619^(4*320) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 23
Running N+1 test using discriminant 43, base 9+sqrt(43)
Calling N-1 BLS with factored part 33.29% and helper 0.14% (100.04% proof)
1-1619^320+1619^(2*320)-1619^(3*320)+1619^(4*320) is prime! (5.9912s+0.0007s)

• leizhou says:

a(21)=Phi[10,1156^256]=1-1156^256+1156^(2*256)-1156^(3*256)+1156^(4*256)

3,137 digits

Found Prime Factors of p-1:
2
3
5
7
11
13
17
89
97
257
47441
1336337
7477121
37642417
2583249857
49521227489
65959705961729
207413006868032513
154186600910808898663635581124287233

Proven PRP by OpenPFGW:

Primality testing 1-1156^256+1156^(2*256)-1156^(3*256)+1156^(4*256) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N+1 test using discriminant 29, base 10+sqrt(29)
Calling N-1 BLS with factored part 29.08% and helper 0.36% (87.62% proof)
1-1156^256+1156^(2*256)-1156^(3*256)+1156^(4*256) is Fermat and Lucas PRP! (2.3154s+0.0003s)

CHG proof screen output:
(13:52) gp > \r examples\CHG
realprecision = 16000 significant digits

Welcome to the CHG primality prover!
————————————

Input file is: Y:\prime\pfgw3.3.2\phis\CHG\256\GGF_n5_256.in
Certificate file is: Y:\prime\pfgw3.3.2\phis\CHG\256\GGF_n5_256.out
Found values of n, F and G.
Number to be tested has 3137 digits.
Modulus has 907 digits.
Modulus is 28.89626768% of n.

NOTICE: This program assumes that n has passed
a BLS PRP-test with n, F, and G as given. If
not, then any results will be invalid!

Square test passed for F >> G. Using modified right endpoint.

Search for factors congruent to 1.
Running CHG with h = 6, u = 2. Right endpoint has 418 digits.
Done! Time elapsed: 193187ms.
Running CHG with h = 6, u = 2. Right endpoint has 370 digits.
Done! Time elapsed: 183610ms.
Running CHG with h = 5, u = 1. Right endpoint has 275 digits.
Done! Time elapsed: 33000ms.
Running CHG with h = 5, u = 1. Right endpoint has 162 digits.
Done! Time elapsed: 16546ms.
A certificate has been saved to the file: Y:\prime\pfgw3.3.2\phis\CHG\256\GGF_n
5_256.out

Running David Broadhurst’s verifier on the saved certificate…

Testing a PRP called “Y:\prime\pfgw3.3.2\phis\CHG\256\GGF_n5_256.in”.

Pol[1, 1] with [h, u]=[4, 1] has ratio=7.77298486 E-245 at X, ratio=5.79941090 E
-339 at Y, witness=61.
Pol[2, 1] with [h, u]=[4, 1] has ratio=5.79941090 E-339 at X, ratio=1.796640949
E-113 at Y, witness=61.
Pol[3, 1] with [h, u]=[6, 2] has ratio=1.314808601 E-106 at X, ratio=9.61072818
E-191 at Y, witness=2.
Pol[4, 1] with [h, u]=[6, 2] has ratio=1.894588607 E-97 at X, ratio=1.894588607
E-97 at Y, witness=2.

Validated in 1 sec.

Congratulations! n is prime!
Goodbye!

CHG certificate: here

• leizhou says:

a(20)=Phi[10,911^250]=1-911^250+911^(2*250)-911^(3*250)+911^(4*250)

2,960 digits

Found Prime Factors of a(20)-1 (GGF_n5_250.helper):
2
3
5
7
11
13
19
251
401
701
761
911
929
1249
1601
1721
21601
27751
45751
217201
296801
5992751
17884211
399775501
129053101501
14177283116650751
224799415189780702751
9727681571205965505101
225404233715206382855642201
687710207363280131260208541153901
1361593766076481797345010718790317173721

Primality testing 1-911^250+911^(2*250)-911^(3*250)+911^(4*250) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N+1 test using discriminant 53, base 6+sqrt(53)
Calling N-1 BLS with factored part 33.79% and helper 0.01% (101.38% proof)
1-911^250+911^(2*250)-911^(3*250)+911^(4*250) is prime! (2.9712s+0.0006s)

• leizhou says:

a(19)=Phi[10,521^200]=1-521^200+521^(2*200)-521^(3*200)+521^(4*200)

2,174 digits

Found Prime Factors of a(19)-1 (GGF_n5_200.helper):
2
3
5
11
13
29
41
61
73
101
113
151
181
461
521
701
1301
1931
4001
9281
9901
12641
17761
75571
135721
307201
671701
1177801
1398521
4464451
4466009
13976701
38083411
1587869681
21309676961
50977416241
265657428737
630005745401
13700492067601
1164303475151101
12051659678309401
19443503877299101
38880452674681601
10082608501590846691648561
180563093712392283737330753
919011617386572718190834546555970401
131910046294478872520745897311794873248701
21086259227221421906234660707211504693097860101

Primality testing 1-521^200+521^(2*200)-521^(3*200)+521^(4*200) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N-1 test using base 23
Running N+1 test using discriminant 37, base 12+sqrt(37)
Calling N-1 BLS with factored part 44.38% and helper 0.54% (133.68% proof)
1-521^200+521^(2*200)-521^(3*200)+521^(4*200) is prime! (1.6821s+0.0005s)

• leizhou says:

a(18)=Phi[10,2^160]=1-2^160+2^(2*160)-2^(3*160)+2^(4*160)

193 digits

Equals to a(14)

• leizhou says:

a(17)=Phi[10,56^128]=1-56^128+56^(2*128)-56^(3*128)+56^(4*128)

896 digits

Prime Factors of a(17)-1:
GGF_n5_128.helper:
2
7
5
11
17
3137
3329
4289
9834497
12324161
81227777
112790017
422229601
461386369
272743988641
5689253622001
478998073521217
9204182701393835713

Primality testing 1-56^128+56^(2*128)-56^(3*128)+56^(4*128) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N+1 test using discriminant 37, base 12+sqrt(37)
Calling N-1 BLS with factored part 38.45% and helper 0.50% (115.94% proof)
1-56^128+56^(2*128)-56^(3*128)+56^(4*128) is prime! (0.2256s+0.0004s)

• leizhou says:

a(16)=Phi[10,232^125]

888 digits

Prime Factors of a(16)-1:
GGF_n5_125.helper:

2
3
5
7
11
29
151
251
281
2153
2531
4091
4751
13001
97001
1449001
2435201
4893001
17304123044101
554942437101822882067505427990915978151
170205694806910699747165388331619297890329020\
3044796270355772478800051470158947227462501

Primality testing 1-232^125+232^(2*125)-232^(3*125)+232^(4*125) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Running N-1 test using base 17
Running N-1 test using base 19
Running N+1 test using discriminant 43, base 2+sqrt(43)
Calling N-1 BLS with factored part 41.45% and helper 0.56% (124.92% proof)
1-232^125+232^(2*125)-232^(3*125)+232^(4*125) is prime! (0.5028s+0.0007s)

• leizhou says:

a(15)=Phi[10,65^100]

544 digits

Prime Factors of a(15)-1:
GGF_n5_100.helper:

2
3
5
11
13
17
101
113
151
401
577
971
2113
2741
18671
129281
174061
8455217
82825201
116223791261
16217230744901
1349360331672401
2737143295980601
44870895496379101
3244699302048456001
12003010477294235790586577227997351
24576399296177435607133227397021121
26746810111375495742736091979976297270301

Primality testing 1-65^100+65^(2*100)-65^(3*100)+65^(4*100) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N-1 test using base 29
Running N-1 test using base 37
Running N-1 test using base 43
Running N-1 test using base 47
Running N+1 test using discriminant 67, base 3+sqrt(67)
Calling N-1 BLS with factored part 60.38% and helper 0.91% (182.06% proof)
1-65^100+65^(2*100)-65^(3*100)+65^(4*100) is prime! (0.3322s+0.0005s)

• leizhou says:

a(14)=Phi[10,4^80]

193 digits

Phi[160,2]
3* 5^2* 11* 17* 31* 41* 257* 61681* 65537* 414721* 4278255361* 44479210368001

Phi[128,2]=1+2^64 274177* 67280421310721
Phi[640,2]=1-2^64+2^128-2^192+2^256
286721* 446960641* 96645260801* 3442404051886487041* 2715862005931406599419575483412481

GGF_n5_80.helper
2
3
5
11
17
31
41
257
61681
65537
274177
286721
414721
446960641
4278255361
96645260801
44479210368001
67280421310721
3442404051886487041
2715862005931406599419575483412481

Primality testing 1-4^80+4^(2*80)-4^(3*80)+4^(4*80) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N-1 test using base 23
Running N-1 test using base 29
Running N-1 test using base 37
Running N+1 test using discriminant 47, base 1+sqrt(47)
Calling N-1 BLS with factored part 100.00% and helper 0.78% (301.25% proof)
1-4^80+4^(2*80)-4^(3*80)+4^(4*80) is prime! (0.0495s+0.0004s)

• leizhou says:

a(13)=Phi[10,373^64]

494 digits

From Factorisation
372 2^2 * 3 * 31
Phi[2,373]=374 2 * 11 * 17
Phi[4,373]=139130 2 * 5 * 13913
Phi[8,373]=19356878642 2 * 54217 * 178513
Phi[16,373]=374688750722402006882
2 * 187344375361201003441
Phi[32,373]=140391659917914310433331237045696371348162
2 * 371873 * 19959073 * 9457498897039913538990121889
Phi[64,373]=19709818174507307565719443829033131863966386128853447415328085964703029568678081922
2 * 193 * 19009 * 14557889 * 78517500128487001623811457 * 2350017925150631169169033327337241144485761

GGF_n5_64.helper
2
3
5
11
17
31
193
373
13913
19009
54217
178513
371873
14557889
19959073
38389249
55615998804579329
187344375361201003441
78517500128487001623811457
9457498897039913538990121889
2350017925150631169169033327337241144485761

Primality testing 1-373^64+373^(2*64)-373^(3*64)+373^(4*64) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 23
Running N-1 test using base 41
Running N-1 test using base 47
Running N+1 test using discriminant 61, base 12+sqrt(61)
Calling N-1 BLS with factored part 53.73% and helper 1.28% (162.51% proof)
1-373^64+373^(2*64)-373^(3*64)+373^(4*64) is prime! (0.1974s+0.0003s)

• leizhou says:

a(12)=Phi[10,5^50]

622301527786114170714406405378012417052532895248031\
358691463745029389305888319156836996164700086470525\
36332466843305155634880065917968750001

140 digits

Equals to a(9)

• leizhou says:

a(11)=Phi[10,14^40]

24015172390093493813079336635335038509051217937081\
73226855310529779167285337174591336512532534092427\
52395961678355361368665765608412685540200505075790\
1238577755035196976617032043724801

184 digits

Primality testing 1-14^40+14^(2*40)-14^(3*40)+14^(4*40) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N-1 test using base 19
Running N-1 test using base 23
Running N+1 test using discriminant 47, base 1+sqrt(47)
Calling N-1 BLS with factored part 46.63% and helper 2.13% (142.53% proof)
1-14^40+14^(2*40)-14^(3*40)+14^(4*40) is prime! (0.0333s+0.0004s)