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Old 2014-10-12, 09:10   #1
primus
 
Jul 2014
Montenegro

2×13 Posts
Default 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^8-8*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 .
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