Chebyshev polynomials of the first kind and primality testing
Conjecture 41
Let n be a natural number greater than two . Let r be the smallest odd prime number such that r \nmid n and n^2 \not\equiv 1 \pmod r . Let T_n(x) be Chebyshev polynomial of the first kind , then n is a prime number if and only if T_n(x) \equiv x^n \pmod {x^r-1,n} .
See Chebyshev polynomials of the first kind and primality testing and Conjecture 41 of Peđa Terzić.
2021, Olexa Bilaniuk & Dendi Suhubdy
Hey @MartinMystery! This was inspired by your algorithm and post on MathOverflow.net.
Would you think we could publish this?
@tsirif @obilaniu Agrawal, Kayal and Saxena suggest a variant of their algorithm which would run in {\displaystyle {\tilde {O}}(\log(n)^{3})} {\displaystyle {\tilde {O}}(\log(n)^{3})} if Agrawal's conjecture is true; however, a heuristic argument by Hendrik Lenstra and Carl Pomerance suggests that it is probably false.
Popovych made a note to Agrawal's conjecture here https://eprint.iacr.org/2009/008.pdf, I wonder if we could modify the code to adjust to Popovych's conjecture?
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