“00001:Prime[5]:11+/-6*1=5+17” “00002:Prime[8]:19+/-6*2=7+31” “00003:Prime[18]:61+/-6*3=43+79” “00004:Prime[14]:43+/-6*4=19+67” “00005:Prime[25]:97+/-6*5=67+127” “00006:Prime[38]:163+/-6*6=127+199” “00007:Prime[43]:191+/-6*7=149+233” “00008:Prime[50]:229+/-6*8=181+277” “00009:Prime[61]:283+/-6*9=229+337” “00010:Prime[48]:223+/-6*10=163+283” “00011:Prime[132]:743+/-6*11=677+809” “00012:Prime[167]:991+/-6*12=919+1063” “00013:Prime[100]:541+/-6*13=463+619” “00014:Prime[88]:457+/-6*14=373+541” “00015:Prime[151]:877+/-6*15=787+967” “00016:Prime[217]:1327+/-6*16=1231+1423” “00017:Prime[176]:1049+/-6*17=947+1151” “00018:Prime[216]:1321+/-6*18=1213+1429” “00019:Prime[270]:1733+/-6*19=1619+1847” “00020:Prime[214]:1307+/-6*20=1187+1427” “00021:Prime[300]:1987+/-6*21=1861+2113” “00022:Prime[785]:6011+/-6*22=5879+6143” “00023:Prime[429]:2971+/-6*23=2833+3109” “00024:Prime[687]:5153+/-6*24=5009+5297” “00025:Prime[308]:2029+/-6*25=1879+2179” “00026:Prime[1083]:8693+/-6*26=8537+8849” “00027:Prime[374]:2551+/-6*27=2389+2713” “00028:Prime[644]:4789+/-6*28=4621+4957” “00029:Prime[713]:5407+/-6*29=5233+5581” “00030:Prime[320]:2129+/-6*30=1949+2309” “00031:Prime[840]:6473+/-6*31=6287+6659” “00032:Prime[608]:4481+/-6*32=4289+4673” “00033:Prime[654]:4889+/-6*33=4691+5087” “00034:Prime[577]:4217+/-6*34=4013+4421” “00035:Prime[1005]:7951+/-6*35=7741+8161” “00036:Prime[1409]:11743+/-6*36=11527+11959” “00037:Prime[1631]:13789+/-6*37=13567+14011” “00038:Prime[1215]:9851+/-6*38=9623+10079” “00039:Prime[928]:7253+/-6*39=7019+7487” “00040:Prime[1386]:11491+/-6*40=11251+11731” “00041:Prime[2304]:20393+/-6*41=20147+20639” “00042:Prime[1984]:17231+/-6*42=16979+17483” “00043:Prime[1203]:9749+/-6*43=9491+10007” “00044:Prime[2336]:20747+/-6*44=20483+21011” “00045:Prime[853]:6599+/-6*45=6329+6869” “00046:Prime[1638]:13873+/-6*46=13597+14149” “00047:Prime[1899]:16369+/-6*47=16087+16651” “00048:Prime[1806]:15461+/-6*48=15173+15749” “00049:Prime[1974]:17123+/-6*49=16829+17417” “00050:Prime[1594]:13451+/-6*50=13151+13751” “00051:Prime[1228]:9967+/-6*51=9661+10273” “00052:Prime[2958]:26959+/-6*52=26647+27271” “00053:Prime[2371]:21089+/-6*53=20771+21407” “00054:Prime[4376]:41863+/-6*54=41539+42187” “00055:Prime[2999]:27437+/-6*55=27107+27767” […]

Using Primo, 274^2311 – 83 is proven a prime number. Define this as p[1]=a[1]-b[1], while a[1]=274^2311 and b[1]=83. p[2]=16236*(a[1]^2-b[1]^2)-1 =16236*274^4622 – 111849805 is proven prime using pfgw: pfgw -h”p[1]” -tp “p[2]” Keep going in this way, it is obtained: p[3]=2^9249*3^5*5*7*11^2*13^2*41^2*137^9244 – (3643*121875747021497257) p[4]=2^18502*3^10*5^2*7^2*11^4*13^4*41^4*137^18488*1223 – ((3643*121875747021497257)^2*19568-1) p[5]=2^37007*3^20*35^4*5863^8*137^36976*953*1223^2 – (((3643*121875747021497257)^2*19568-1)^2*7624+1) p[6]=2^74015*3^42*35^8*5863^16*137^73952*953^2*1223^4*15217 – ((((3643*121875747021497257)^2*19568-1)^2*7624+1)^2*273906-1) Certificate will be posted […]

Define: p[k,i]=ABS[1+2*n[k,i]*p[k-1,1]*p[k-1,2]],n[k,1] is the integer with minimum ABS[n[k,1]] that makes p[k,1] a prime number, and n[k,2] is the integer with second minimum ABS[n[k,2]] that makes p[k,2] a prime number The primality of p[k,i] can be proven using Brillhart – Lehmer – Selfridge algorithm recursively by using p[k-1,1] and p[k-1,2] as helper since n is a […]

Define p(0)=1; finding the smallest General Fermat prime in the form p(n+1)[m]=(2*m*p(n))^2+1, m is positive integer: p(1)[1]=(2*p(0))^2+1=5; p(2)[1]=(2*p(1))^2+1=101; p(3)[5]=(2*5*p(2))^2+1=1020101; p(4)[48]=(2*48*p(3))^2+1=((1020101)*96)^2+1; p(5)[1]=(2*p(4))^2+1=((((1020101)*96)^2+1)*2)^2+1; p(6)[30]=(2*30*p(5))^2+1=((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1; p(7)[85]=(2*85*p(6))^2+1=((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1; p(8)[935]=(2*935*p(7))^2+1=((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1; p(9)[528]=(2*528*p(8))^2+1=((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1; p(10)[2505]=(2*2505*p(9))^2+1=((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1; p(11)[840]=(2*840*p(10))^2+1=((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1; p(12)[1190]=(2*1190*p(11))^2+1=((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1; p(13)[29382]=(2*29382*p(12))^2+1=((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1; p(14)[25176]=(2*25176*p(13))^2+1=((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1; p(15)[12685]=(2*12685*p(14))^2+1=((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1; p(16)[67852]=(2*67852*p(15))^2+1=((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1; p(17)[299549]=(2*299549*p(16))^2+1=((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1; p(18)[62406]=(2*62406*p(17))^2+1=((((((((((((((((((((((((((((97929696^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1)*124812)^2+1; p(4) has database ID 96548 in The List of Largest Known Primes Home Page. The direct link is HERE. These primes […]

Define p(0)=12^9*5^5^5+7; p(1)[m=466;n=78]=466*((78*(12^9*5^3125+7))^3+1)+1; p(2)[m=6470;n=884]=6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1; p(3)[m=278822;n=33410]=278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1; p(4)[m=145950;n=46953]=145950*((46953*( 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1; p(4) has database ID 96540 in The List of Largest Known Primes Home Page. The direct link is HERE. The kernel 12^9*5^5^5+1 is proven by Primo. The certificate is in the first reply of this post. The recursive primes are proven using OpenPFGW, by the command pfgw -t […]

1) For any number N=Sigma(p_i), i=1..k, p_i are prime factors of N with any prime number cp, cp is not factor of N, there is: Mod(cp^LCM(p_i-1, i=1..k), N)=1. for some case, Mod(cp^(LCM(p_i-1, i=1..k)/2), N)= +/-1 Special: when N is prime, Mod(cp^(N-1), N)=1 For example: In[3]:= FactorInteger[66855224152] Out[3]= {{2, 3}, {19, 1}, {1549, 1}, {283949, 1}} […]

Define p(1)=2 p(2)[m=-1; +1] = p(1)*(p(1)-1)+1 = 3 p(3)[m=-1; -1] = p(2)*(p(2)-1)-1 = 5 p(4)[m=-1; -1] = p(3)*(p(3)-1)-1 = 19 p(5)[m=+1; -1] = p(4)*(p(4)+1)-1 = 379 p(6)[m=-1; -1] = p(5)*(p(5)-1)-1 = 143261 p(7)[m=-11; -1] = p(6)*(p(6)-11)-1 = 20522138249 p(8)[m=-11; +1] = p(7)*(p(7)-11)+1 = 421158158085325265263 p(9)[m=-13; +1] = p(8)*(p(8)-13)+1 = 421158158085325265256.5^2-165/4 p(10)[m=+59; -1] = p(9)*(p(9)+59)-1 = […]

Define p(0)=1; p(1)[m=2;n=2]=2*((2*p(0))^3+1)+1=19; p(2)[m=6;n=4]=6*((4*p(1))^3+1)+1=2633863; p(3)[m=14;n=1]=14*((1*2633863)^3+1)+1=14*2633863^3+15; p(4)[m=354;n=74]=354*((74*(14*2633863^3+15 ))^3+1)+1 =354*(1036*2633863^3+1110)^3+355; p(5)[m=155;n=115]=155*((115*(354*(1036*2633863^3+1110)^3+355 ))^3+1)+1 =155*(40710*(1036*2633863^3+1110)^3+40825)^3+156; p(6)[m=146;n=629]=146*((629*(155*(40710*(1036*2633863^3+1110)^3+40825)^3+156 ))^3+1)+1 =146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147; p(7)[m=440;n=1754]=440*((1754*(146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147 ))^3+1)+1 =440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441; p(8)[m=8385;n=185]=8385*((185*(440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441 ))^3+1)+1 =8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386; p(9)[m=16182;n=2988]=16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386 ))^3+1)+1; p(10)[m=79194;n=97326]=79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1; p(11)=[m=232497;n=176845]=232497*((176845*(79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1))^3+1)+1; p(11) has database ID 94439 in The List of Largest Known Primes Home Page. The direct link is HERE. These primes are recursively proven using OpenPFGW, by the command pfgw -t […]