20220110, 11:51  #1 
Feb 2004
France
927_{10} 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^22). 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 
20220111, 21:40  #2 
Feb 2004
France
3^{2}·103 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,q1,S=Mod(S^22,w));s1=lift(Mod(S59,w));s2=lift(Mod(S5+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 
20220115, 14:44  #3 
Feb 2017
Nowhere
1011010001001_{2} 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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