20220110, 11:51  #1 
Feb 2004
France
929 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
1110100001_{2} 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
13·461 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.

20220612, 20:37  #4 
Mar 2021
France
101001_{2} Posts 
I don't know if it's related to this topic but I made some probable primality test for numbers of the forum (a^p1)/(a1) and (a^p+1)/(a+1) using Chebyshev polynomials :
but the test isn't perfect and there are some conditions to apply :  a must not be a perfect power otherwise you can get false positive.  p must be a prime number >2 otherwise you can "break" the primality test and get false positive Let Let the sequence where is the Chebyshev's polynomial of the first kind with is prime if or You can found the test here : https://sagecell.sagemath.org/?z=eJx...yLjgUAARUAuQ== For Let the sequence where is the Chebyshev's polynomial of the first kind with is prime if or You can found the test here : https://sagecell.sagemath.org/?z=eJx...yLjgUAARUAuQ== For the moment I didn't get a counterexample If you found one please tell me. EDIT : (Second test fails for (a^3+1)/(a+1) apparently where a = 5 and 7) Last fiddled with by kijinSeija on 20220612 at 21:20 
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