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 2022-01-10, 11:51 #1 T.Rex     Feb 2004 France 39F16 Posts Chebyshev polynomials and higher order Lucas Lehmer algorithm by Kok Seng Chu Hi, I've found this recent (2021, October 3rd) paper named "CHEBYSHEV POLYNOMIALS AND HIGHER ORDER LUCAS LEHMER ALGORITHM", by KOK SENG CHUA, based on previous work by Pedja Terzi´c, and talking about the necessity part. This looks very interesting to me, since it provides a search about generalized Mersennes and the LLT (x^2-2). However, I've not spent yet enough time to read it, and it's not easy for me to understand it. So, I'd like to get comments from true mathematicians. https://arxiv.org/pdf/2010.02677.pdf Regards
 2022-01-11, 21:40 #2 T.Rex     Feb 2004 France 11100111112 Posts The formula for Wagstaff numbers seems OK. With Pari/gp . Code: t(q)={w=(2^q+1)/3;S0=4;print("w: ",w);S=S0;for(i=1,q-1,S=Mod(S^2-2,w));s1=lift(Mod(S-5-9,w));s2=lift(Mod(S-5+9,w));print(s1," ",s2)} ? t(11) w: 683 665 0 ? t(13) w: 2731 0 18 ? t(17) w: 43691 43673 0 ? t(19) w: 174763 0 18 ? t(23) w: 2796203 2796185 0 ? t(29) w: 178956971 Not prime 59834419 59834437 ? t(31) w: 715827883 0 18 ? t(37) w: 45812984491 Not prime 24875527143 24875527161 ? t(41) w: 733007751851 Not prime 634893124730 634893124748 ? t(43) w: 2932031007403 0 18 ? t(61) w: 768614336404564651 0 18
 2022-01-15, 14:44 #3 Dr Sardonicus     Feb 2017 Nowhere 32×641 Posts The sufficiency result (Lemma 2.1) requires a complete factorization of Q + 1 or Q - 1 in order to prove that Q is prime.

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