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