My work, my view, and my imagination
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 […]
Define p(0)=1; p(1)=2*P(0)^2+1=3; p(2)=2*p(1)^2-1=17; p(3)=2*p(2)^2-1=577; p(4)=2*p(3)^2-1=665857; p(5)=20*p(4)^2-1; p(6)=2*p(5)^2-1; p(7)=28*p(6)^2+1; p(8)=182*p(7)^2-1; p(9)=272*p(8)^2-1; p(10)=540*p(9)^2+1; p(11)=162*p(10)^2+1; p(12)=1002*p(11)^2+1; p(13)=112*p(12)^2+1; p(14)=306*p(13)^2-1; p(15)=1752*p(14)^2+1; p(16)=20564*p(15)^2-1; p(17)=135236*p(16)^2-1; p(18)=547952*p(17)^2-1; p(19)=282904*p(18)^2+1; p(19)=282904*(547952*(135236*(20564*(1752*(306*(112*(1002*(162*(540*(272*(182*(28*(2*8867310888979^2-1)^2+1)^2-1)^2-1)^2+1)^2+1)^2+1)^2+1)^2-1)^2+1)^2-1)^2-1)^2-1)^2+1 has database ID 94235 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 (or tp) -h”p(k)” […]
Define p(0)=2; p(1)=2*(2-1)+1 = 3 is prime with m(1)=1; (+1) p(2)=3*(3-1)-1 = 5 is prime with m(2)=1; (-1) p(3)=5*(5-1)-1 = 19 is prime with m(3)=1; (-1) p(4)=19*(19-7)-1 = 227 is prime with m(4)=7; (-1) p(5)=227*(227-3)+1 = 50849 is prime with m(5)=3; (+1) p(6)=50849*(50849-29)-1 = 2584146179 is prime with m(6)=29; (-1) p(7)=2584146179*(2584146179-19)-1 = 6677811425341522639 is prime […]
Take any three primes, say p[1,0], p[2,0], and p[3,0]. Define: p[i,j]=ABS[1+2*n[i,j]*p[(i+1) mod 3,j-1]*p[(i+2) mod 3,j-1]],n is the integer with minimum ABS[n] that makes p[i,j] a prime number. The primality of p[i,j] can be proven using Brillhart – Lehmer – Selfridge algorithm recursively by using p[(i+1) mod 3,j-1] and p[(i+2) mod 3,j-1] as helper since n […]
Brillhart – Lehmer – Selfridge algorithm provides a general primality proving method as long as you can factor P+1 or P-1. Therefore, for any prime number, when P+1 or P-1 get fully factored, the primality of any factors of P+1 or P-1 can also be proven by the same algorithm recursively. For example, prime number […]
18762*2*(41968149*2^23209+1)*(97254741*2^21701+1)*(12450795*2^11213+1)*(20092671*2^11213+1)*(28115529*2^11213+1)*(43998375*2^11213+1)*(99539031*2^11213+1)*(14024781*2^9941+1)*(26534229*2^9941+1)*(78981231*2^9689+1)*(57506259*2^96 89+1)*(24975705*2^9689+1)*(7770399*2^9689+1)*(99738639*2^4423+1)*(97478625*2^4423+1)*(96783729*2^4423+1)*(86445039*2^4423+1)*(86022459*2^4423+1)*(82860819*2^4423+1)*(82852119*2^4423+1)*(54983739*2^4423+1)*(37753119*2^4423+1)*(264 48255*2^4423+1)*(26055441*2^4423+1)*(24439281*2^4423+1)*(22820631*2^4423+1)*(17436999*2^4423+1)*(14395845*2^4423+1)*(12058929*2^4423+1)*(7288671*2^4423+1)*(96317565*2^4253+1)*(82195221*2^4253+1)*(77293269*2^4253+1)*(71596305*2^42 53+1)*(57309375*2^4253+1)*(56921445*2^4253+1)*(34719825*2^4253+1)*(16894545*2^4253+1)*(1792431*2^4253+1)*(97688691*2^3217+1)*(91257465*2^3217+1)*(91090419*2^3217+1)*(86093541*2^3217+1)*(85877469*2^3217+1)*(84069831*2^3217+1)*(795 33441*2^3217+1)*(78684369*2^3217+1)*(76317555*2^3217+1)*(76037751*2^3217+1)*(72552339*2^3217+1)*(68274339*2^3217+1)*(58155531*2^3217+1)*(53160021*2^3217+1)*(48317421*2^3217+1)*(44497059*2^3217+1)*(42114315*2^3217+1)*(41096715*2^3 217+1)*(39648561*2^3217+1)*(39327861*2^3217+1)*(34314159*2^3217+1)*(29652849*2^3217+1)*(24763389*2^3217+1)*(24430989*2^3217+1)*(23748699*2^3217+1)*(20007255*2^3217+1)*(18596781*2^3217+1)*(13656495*2^3217+1)*(11109435*2^3217+1)*(8 869755*2^3217+1)*(6894645*2^3217+1)*(5209209*2^3217+1)*(2851731*2^3217+1)*(96288201*2^2281+1)*(95937261*2^2281+1)*(95933985*2^2281+1)*(94364775*2^2281+1)*(92893239*2^2281+1)*(92748381*2^2281+1)*(90589905*2^2281+1)*(87734841*2^228 1+1)*(86566965*2^2281+1)*(86258499*2^2281+1)*(85734651*2^2281+1)*(84715839*2^2281+1)*(81532725*2^2281+1)*(79884849*2^2281+1)*(79081275*2^2281+1)*(76694835*2^2281+1)*(72081945*2^2281+1)*(72017001*2^2281+1)*(71253915*2^2281+1)*(688 93959*2^2281+1)*(68085225*2^2281+1)*(65410995*2^2281+1)*(63626145*2^2281+1)*(60039321*2^2281+1)*(56002311*2^2281+1)*(54061581*2^2281+1)*(53434575*2^2281+1)*(49409151*2^2281+1)*(41855025*2^2281+1)*(40910079*2^2281+1)*(40303905*2^2 281+1)*(40129041*2^2281+1)*(40018299*2^2281+1)*(31393209*2^2281+1)*(30810039*2^2281+1)*(27754761*2^2281+1)*(22185849*2^2281+1)*(20176311*2^2281+1)*(16289589*2^2281+1)*(14164635*2^2281+1)*(14147229*2^2281+1)*(11236485*2^2281+1)*(1 0213875*2^2281+1)*(4128945*2^2281+1)*(4039659*2^2281+1)*(4018065*2^2281+1)*(3928155*2^2281+1)*(3015531*2^2281+1)*(98028735*2^2203+1)*(97108935*2^2203+1)*(96046179*2^2203+1)*(95560221*2^2203+1)*(94884285*2^2203+1)*(94506579*2^2203 +1)*(91113195*2^2203+1)*(89055891*2^2203+1)*(88513845*2^2203+1)*(88293609*2^2203+1)*(87081549*2^2203+1)*(84254751*2^2203+1)*(78451869*2^2203+1)*(77693889*2^2203+1)*(76733511*2^2203+1)*(76650939*2^2203+1)*(76129479*2^2203+1)*(7458 5799*2^2203+1)+1 Proof: $ more cp1.cert $ ./pfgw -t -h”helper” cp1 PFGW Version 3.3.2.20100216.x86_Dev [GWNUM 25.13] Resuming input file cp1 at line 2 Primality testing 18762*2*(41968149*2^23209+1)*(97254741*2^21701+1)*(12450795*2^11213+1)*(20092671*2^11213+1)*(28115529*2^11213+1)*(43998375*2^11213+1)*(99539031*2^11213+1)*(14024781*2^9941+1)*(26534229*2^9941+1)*(78981231*2^9689+1)*(57506259*2^96 89+1)*(24975705*2^9689+1)*(7770399*2^9689+1)*(99738639*2^4423+1)*(97478625*2^4423+1)*(96783729*2^4423+1)*(86445039*2^4423+1)*(86022459*2^4423+1)*(82860819*2^4423+1)*(82852119*2^4423+1)*(54983739*2^4423+1)*(37753119*2^4423+1)*(264 48255*2^4423+1)*(26055441*2^4423+1)*(24439281*2^4423+1)*(22820631*2^4423+1)*(17436999*2^4423+1)*(14395845*2^4423+1)*(12058929*2^4423+1)*(7288671*2^4423+1)*(96317565*2^4253+1)*(82195221*2^4253+1)*(77293269*2^4253+1)*(71596305*2^42 53+1)*(57309375*2^4253+1)*(56921445*2^4253+1)*(34719825*2^4253+1)*(16894545*2^4253+1)*(1792431*2^4253+1)*(97688691*2^3217+1)*(91257465*2^3217+1)*(91090419*2^3217+1)*(86093541*2^3217+1)*(85877469*2^3217+1)*(84069831*2^3217+1)*(795 33441*2^3217+1)*(78684369*2^3217+1)*(76317555*2^3217+1)*(76037751*2^3217+1)*(72552339*2^3217+1)*(68274339*2^3217+1)*(58155531*2^3217+1)*(53160021*2^3217+1)*(48317421*2^3217+1)*(44497059*2^3217+1)*(42114315*2^3217+1)*(41096715*2^3 217+1)*(39648561*2^3217+1)*(39327861*2^3217+1)*(34314159*2^3217+1)*(29652849*2^3217+1)*(24763389*2^3217+1)*(24430989*2^3217+1)*(23748699*2^3217+1)*(20007255*2^3217+1)*(18596781*2^3217+1)*(13656495*2^3217+1)*(11109435*2^3217+1)*(8 869755*2^3217+1)*(6894645*2^3217+1)*(5209209*2^3217+1)*(2851731*2^3217+1)*(96288201*2^2281+1)*(95937261*2^2281+1)*(95933985*2^2281+1)*(94364775*2^2281+1)*(92893239*2^2281+1)*(92748381*2^2281+1)*(90589905*2^2281+1)*(87734841*2^228 1+1)*(86566965*2^2281+1)*(86258499*2^2281+1)*(85734651*2^2281+1)*(84715839*2^2281+1)*(81532725*2^2281+1)*(79884849*2^2281+1)*(79081275*2^2281+1)*(76694835*2^2281+1)*(72081945*2^2281+1)*(72017001*2^2281+1)*(71253915*2^2281+1)*(688 93959*2^2281+1)*(68085225*2^2281+1)*(65410995*2^2281+1)*(63626145*2^2281+1)*(60039321*2^2281+1)*(56002311*2^2281+1)*(54061581*2^2281+1)*(53434575*2^2281+1)*(49409151*2^2281+1)*(41855025*2^2281+1)*(40910079*2^2281+1)*(40303905*2^2 281+1)*(40129041*2^2281+1)*(40018299*2^2281+1)*(31393209*2^2281+1)*(30810039*2^2281+1)*(27754761*2^2281+1)*(22185849*2^2281+1)*(20176311*2^2281+1)*(16289589*2^2281+1)*(14164635*2^2281+1)*(14147229*2^2281+1)*(11236485*2^2281+1)*(1 0213875*2^2281+1)*(4128945*2^2281+1)*(4039659*2^2281+1)*(4018065*2^2281+1)*(3928155*2^2281+1)*(3015531*2^2281+1)*(98028735*2^2203+1)*(97108935*2^2203+1)*(96046179*2^2203+1)*(95560221*2^2203+1)*(94884285*2^2203+1)*(94506579*2^2203 +1)*(91113195*2^2203+1)*(89055891*2^2203+1)*(88513845*2^2203+1)*(88293609*2^2203+1)*(87081549*2^2203+1)*(84254751*2^2203+1)*(78451869*2^2203+1)*(77693889*2^2203+1)*(76733511*2^2203+1)*(76650939*2^2203+1)*(76129479*2^2203+1)*(7458 5799*2^2203+1)+1 [N-1, Brillhart-Lehmer-Selfridge] Reading factors from helper […]
Generalize Woodall: 3*2^3-1 5*4^5-1 10*9^10-1 11*10^11-1 18*17^18-1 127*126^127-1 286*285^286-1 560*559^560-1 1025*2^10250-1 39144*39143^39144-1 Generalized Cullen 2*1^2+1 621*620^621+1 41556*41555^41556+1 tested up to n=50k
4348099×2^2976221-1 A Riesel companion of Merssenne Prime M36. Detail please see http://primes.utm.edu/bios/page.php?id=1273