The sequence:<\/p>\n
2, 4, 2, 6, 2, 20, 20, 26, 25, 10,<\/tt><\/p>\n
14, 5, 373, 4, 65, 232, 56, 2, 521, 911,<\/p>\n
1156, 1619, 647, 511, 34, 2336, 2123, 1274, 2866, 951,<\/p>\n
2199, 1353, 4965, 7396, 13513, 3692, 14103, 32275, 2257, 86,<\/p>\n
3928, 2779, 18781, 85835, 820, 16647, 2468, 26677, 1172, 38361,<\/p>\n
40842,<\/p>\n
1-m^k+m^(2*k)-m^(3^k)+m^(4*k) equals Phi(10*k,m), or Phi(10, m^k).<\/tt><\/p>\n
Proofing status:
\na(1)=Phi[10,2^1]: 2 digits, p^2-1 factored prime.
\na(2)=Phi[10,4^2]: 5 digits, p^2-1 factored prime.
\na(3)=Phi[10,2^4]: 5 digits, Equals to a(2)
\na(4)=Phi[10,6^5]: 16 digits, p^2-1 factored prime.
\na(5)=Phi[10,2^8]: 10 digits, p^2-1 factored prime.
\na(6)=Phi[10,20^10]: 53 digits, OpenPFGW 129.07% proof without helper factorization.
\na(7)=Phi[10,20^16]: 84 digits, OpenPFGW 131.88% proof without helper factorization.
\na(8)=Phi[10,26^20]: 114 digits, OpenPFGW 143.88% proof without helper factorization.
\na(9)=Phi[10,25^25]: 140 digits, OpenPFGW 131.03% proof without helper factorization.
\na(10)=Phi[10,10^32]: 128 digits, OpenPFGW 124.71% proof without helper factorization.
\na(11)=Phi[10,14^40]: 184 digits, OpenPFGW 142.53% proof without helper factorization.
\na(12)=Phi[10,5^50]: 140 digits, Equals to a(9).
\na(13)=Phi[10,373^64]: 494 digits, OpenPFGW 162.51% proof with factored part 53.73%.
\na(14)=Phi[10,4^80]: 193 digits, OpenPFGW 301.25% proof with factored part 100.00%.
\na(15)=Phi[10,65^100]: 544 digits, OpenPFGW 182.06% proof with factored part 60.38%.
\na(16)=Phi[10,232^125]: 888 digits, OpenPFGW with factored part 41.45% and helper 0.56% (124.92% proof).
\na(17)=Phi[10,56^128]: 896 digits, OpenPFGW with factored part 38.45% and helper 0.50% (115.94% proof).
\na(18)=Phi[10,2^160]: 193 digits, Equals to a(14).
\na(19)=Phi[10,521^200]: 2,174 digits, OpenPFGW with factored part 44.38% and helper 0.54% (133.68% proof).
\na(20)=Phi[10,911^250]: 2,960 digits, OpenPFGW with factored part 33.79% and helper 0.01% (101.38% proof).
\na(21)=Phi[10,1156^256]: 3,137 digits, CHG proof with factored part 28.90%.
\na(22)=Phi[10,1619^320]: 4,108 digits, OpenPFGW with factored part 33.29% and helper 0.14% (100.04% proof).
\na(23)=Phi[10,647^400]: 4,498 digits, OpenPFGW with factored part 38.43% and helper 0.15% (115.44% proof).
\na(24)=Phi[10,511^500]: 5,417 digits, OpenPFGW with factored part 33.70% and helper 0.07% (101.17% proof).
\na(25)=Phi[10,34^512]: 3,137 digits, Equals to a(21)
\na(26)=Phi[10,2336^625]: 8,422 digits, kp proof with factored part 30.78%.
\na(27)=Phi[10,2123^640]: 8,517 digits, OpenPFGW with factored part 40.68% and helper 0.06% (122.09% proof).
\na(28)=Phi[10,1274^800]: 9,937 digits, kp proof with factored part 31.61%.
\na(29)=Phi[10,2866^1000]: 13,830 digits, CHG proof with factored part 27.77%.
\na(30)=Phi[10,951^1024]: 12,199 digits, CHG proof with factored part 26.95%.
\na(31)=Phi[10,2199^1250]: 16,712 digit, CHG proof with factored part 27.09%.
\na(32)=Phi[10,1353^1280]: 16,033 digits, CHG proof with factored part 28.05%.
\na(33)=Phi[10,4965^1600]: 23,654 digits, CHG proof with factored part 29.10%.
\na(34)=Phi[10,7396^2000]: 30,952 digits, CHG proof with factored part 28.58%.
\n*a(35)=Phi[10,13513^2048]: 33,840 digits, PRP with factored part 26.13%.
\na(36)=Phi[10,3692^2500]: 35,673 digits, CHG proof with factored part 27.69%.
\na(37)=Phi[10,14103^2560]: 42,489 digits, CHG proof factored part 26.40% (chgcertd Type error).
\n*a(38)=Phi[10,32275^3123]: 56,361 digits, PRP with factored part 25.84%.
\na(39)=Phi[10,2257^3200]: 42,926 digits, CHG proof with factored part 27.90%.
\na(40)=Phi[10,86^4000]: 30,952 digits, kp proof with factored part 30.02%.
\na(41)=Phi[10,3928^4096]: 58,887 digits, CHG proof factored part 27.04%.
\na(42)=Phi[10,2779^5000]: 68,878 digits, CHG proof with factored part 26.69%.
\n*a(43)=Phi[10,18781^5120]: 87,526 digits, PRP with factored part 26.34%.
\n*a(44)=Phi[10,85835^6250]: 123,342 digits, PRP with factored part 25.84%.
\na(45)=Phi[10,820^6400]: 74,594 digits, CHG proof with factored part 27.12%.
\n*a(46)=Phi[10,16647^8000]: 135,083 digits, PRP with factored part 26.07%.
\n*a(47)=Phi[10,2468^8192]: 111,161 digits, PRP with factored part 25.28%.
\na(48)=Phi[10,26677^10000]: 177,046 digits, CHG proof with factored part 27.16%.
\n*a(49)=Phi[10,1172^10240]: 125,704 digits, PRP with factored part 25.45%.
\n*a(50)=Phi[10,38361^12500]: 229,195 digits, PRP with factored part 25.32%.
\na(51)=Phi[10,40842^12800]: 236,089 digits, CHG proof with factored part 28.14%.<\/p>\n
Note: Phi[10,4^2]=Phi[10,2^4] since 4=2^2
\nNote: Phi[10,25^25]=Phi[10,5^50] since 25=5^2
\nNote: Phi[10,4^80]=Phi[10,2^160] since 4=2^2
\nNote: Phi[10,1156^256]=Phi[10,34^512] since 1156=34^2
\nNote: Phi[10,7396^2000]=Phi[10,86^4000] since 7396=86^2<\/p>\n
a(1)=Phi[10, 2^1]=11<\/p>\n
a(2)=Phi[10,4^2]=61681<\/p>\n
a(3)=Phi[10,2^4]=61681<\/p>\n
a(4)=Phi[10,6^5]=3655688315536801<\/p>\n
OpenPFGW proof:<\/p>\n
$.\/pfgw\u00a0 -tc -q”1-6^5+6^(2*5)-6^(3*5)+6^(4*5)”<\/p>\n
Primality testing 1-6^5+6^(2*5)-6^(3*5)+6^(4*5) [N-1\/N+1, Brillhart-Lehmer-Selfridge]
\nRunning N-1 test using base 7
\nRunning N-1 test using base 13
\nRunning N-1 test using base 17
\nRunning N+1 test using discriminant 23, base 2+sqrt(23)
\nRunning N+1 test using discriminant 23, base 3+sqrt(23)
\nCalling N+1 BLS with factored part 100.00% and helper 100.00% (403.92% proof)
\n1-6^5+6^(2*5)-6^(3*5)+6^(4*5) is prime! (0.0193s+0.0004s)<\/p>\n
a(5)=Phi[10,2^8]=4278255361<\/p>\n
a(6)=Phi[10,20^10]=10995116277758926258176000104857599999989760000000001<\/p>\n
OpenPFGW proof:<\/p>\n
$ .\/pfgw -tc -q”1-20^10+20^(2*10)-20^(3*10)+20^(4*10)”
\nPFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14]<\/p>\n
Primality testing 1-20^10+20^(2*10)-20^(3*10)+20^(4*10) [N-1\/N+1, Brillhart-Lehmer-Selfridge]
\nRunning N-1 test using base 17
\nRunning N-1 test using base 23
\nRunning N-1 test using base 37
\nRunning N+1 test using discriminant 43, base 3+sqrt(43)
\nCalling N-1 BLS with factored part 40.12% and helper 8.72% (129.07% proof)
\n1-20^10+20^(2*10)-20^(3*10)+20^(4*10) is prime! (0.0086s+0.0006s)<\/p>\n
a(7)=Phi[10,20^16]=18446744073709551615971852502328934400000\\
\n0429496729599999999999344640000000000000001<\/p>\n
OpenPFGW proof:<\/p>\n
$ .\/pfgw -tc -q”1-20^16+20^(2*16)-20^(3*16)+20^(4*16)”
\nPFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14]<\/p>\n
Primality testing 1-20^16+20^(2*16)-20^(3*16)+20^(4*16) [N-1\/N+1, Brillhart-Lehmer-Selfridge]
\nRunning N-1 test using base 29
\nRunning N+1 test using discriminant 37, base 15+sqrt(37)
\nCalling N-1 BLS with factored part 42.03% and helper 5.07% (131.88% proof)
\n1-20^16+20^(2*16)-20^(3*16)+20^(4*16) is prime! (0.0102s+0.0006s)<\/p>\n
a(8)=Phi[10,26^20]=157713125193403080417654809073419921485591\\
\n4988749414250433529072806660344375821341361193668309978857119\\
\n00082176001<\/p>\n
OpenPFGW proof:<\/p>\n
$ .\/pfgw -tc GGF_n5_20
\nPFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14]<\/p>\n
Primality testing 1-26^20+26^(2*20)-26^(3*20)+26^(4*20) [N-1\/N+1, Brillhart-Lehmer-Selfridge]
\nRunning N-1 test using base 17
\nRunning N-1 test using base 23
\nRunning N+1 test using discriminant 43, base 3+sqrt(43)
\nCalling N-1 BLS with factored part 47.34% and helper 1.33% (143.88% proof)
\n1-26^20+26^(2*20)-26^(3*20)+26^(4*20) is prime! (0.0178s+0.0004s)<\/p>\n
a(9)=Phi[10,25^25]=6223015277861141707144064053780124170525328\\
\n95248031358691463745029389305888319156836996164700086470525363\\
\n32466843305155634880065917968750001<\/p>\n
OpenPFGW proof:<\/p>\n
$ .\/pfgw -tc GGF_n5_25
\nPFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14]<\/p>\n
Primality testing 1-25^25+25^(2*25)-25^(3*25)+25^(4*25) [N-1\/N+1, Brillhart-Lehmer-Selfridge]
\nRunning N-1 test using base 13
\nRunning N+1 test using discriminant 19, base 6+sqrt(19)
\nCalling N-1 BLS with factored part 43.53% and helper 0.22% (131.03% proof)
\n1-25^25+25^(2*25)-25^(3*25)+25^(4*25) is prime! (0.0191s+0.0004s)<\/p>\n
a(10)=Phi[10,10^32]=9999999999999999999999999999999900000000000\\
\n000000000000000000000999999999999999999999999999999990000000000\\
\n0000000000000000000001<\/p>\n
OpenPFGW proof:<\/p>\n
$ .\/pfgw -tc GGF_n5_32
\nPFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14]<\/p>\n
Primality testing 1-10^32+10^(2*32)-10^(3*32)+10^(4*32) [N-1\/N+1, Brillhart-Lehmer-Selfridge]
\nRunning N-1 test using base 29
\nRunning N-1 test using base 31
\nRunning N-1 test using base 41
\nRunning N+1 test using discriminant 53, base 13+sqrt(53)
\nCalling N-1 BLS with factored part 40.94% and helper 1.88% (124.71% proof)
\n1-10^32+10^(2*32)-10^(3*32)+10^(4*32) is prime! (0.0205s+0.0004s)<\/p>\n","protected":false},"excerpt":{"rendered":"
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 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,6],"tags":[],"class_list":["post-263","post","type-post","status-publish","format-standard","hentry","category-to-entertain-myself","category-looking-for-a-megaprime","post-blog"],"_links":{"self":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/263","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=263"}],"version-history":[{"count":13,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/263\/revisions"}],"predecessor-version":[{"id":315,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/263\/revisions\/315"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=263"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=263"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}