mersenneforum.org - View Single Post - Conjectured Primality Test for Specific Class of Mersenne Numbers

primus · 2014-10-12, 09:10

Conjecture

Let $M_p=2^p-1$ such that $p$ is prime and $p\equiv 5 \pmod{6}$

Let $S_i=S_{i-1}^8-8\cdot S_{i-1}^6+20\cdot S_{i-1}^4-16 \cdot S_{i-1}^2+2$ with $S_0=4$ , then

$M_p$ is prime iff $S_{(p-2)/3} \equiv 0 \pmod{M_p}$

Maxima Implementations

LL Test

Code:

p:9689;
(s:4,M:2^p-1,
for i from 1 thru (p-2) do (s:mod(s^2-2,M)))$
(if(s=0) then print("prime") else print("composite"));

Conjecture

Code:

p:9689;
(s:4,M:2^p-1,
for i from 1 thru (p-2)/3 do (s:mod(s^8-8*s^6+20*s^4-16*s^2+2,M)))$
(if(s=0) then print("prime") else print("composite"));

Maxima implementation of this modified test is approximately two times faster than Maxima implementation of original Lucas-Lehmer test .

Maybe someone on this forum can prove or disprove this conjecture .

2014-10-12, 09:10	#1
primus Jul 2014 Montenegro 2·13 Posts	Conjectured Primality Test for Specific Class of Mersenne Numbers Conjecture Let $M_p=2^p-1$ such that $p$ is prime and $p\equiv 5 \pmod{6}$ Let $S_i=S_{i-1}^8-8\cdot S_{i-1}^6+20\cdot S_{i-1}^4-16 \cdot S_{i-1}^2+2$ with $S_0=4$ , then $M_p$ is prime iff $S_{(p-2)/3} \equiv 0 \pmod{M_p}$ Maxima Implementations LL Test Code: p:9689; (s:4,M:2^p-1, for i from 1 thru (p-2) do (s:mod(s^2-2,M)))$ (if(s=0) then print("prime") else print("composite")); Conjecture Code: p:9689; (s:4,M:2^p-1, for i from 1 thru (p-2)/3 do (s:mod(s^8-8s^6+20s^4-16*s^2+2,M)))$ (if(s=0) then print("prime") else print("composite")); Maxima implementation of this modified test is approximately two times faster than Maxima implementation of original Lucas-Lehmer test . Maybe someone on this forum can prove or disprove this conjecture .