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 .

axn · 2014-10-12, 09:25

Quote:

Originally Posted by primus

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

This isn't a new test. This is just the LL-test, disguised by using a polynomial which combines 3 iterations of LL into 1. It is only faster because of the interpreted nature of your implementation. I'm betting that the plain version will be faster for larger p (because 3 iterations of s^2-2 should be faster than the deg-8 poly).

EDIT:-

Code:

LL1(p)={my(s=Mod(4,2^p-1)); for(i=1,p-2, s=s^2-2); s==0}
LL2(p)={my(s=Mod(4,2^p-1)); for(i=1,(p-2)/3, s=s^8-8*s^6+20*s^4-16*s^2+2); s==0}
LL3(p)={my(s=Mod(4,2^p-1)); for(i=1,(p-2)/3, s=((s^2-2)^2-2)^2-2); s==0}
LL1(9689)
time = 1,280 ms.
LL2(9689)
time = 3,511 ms.
LL3(9689)
time = 1,276 ms.

In PARI/GP, your version is nearly 3 times slower.

2014-10-12, 09:10	#1
primus Jul 2014 Montenegro 32₈ 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 .

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