Chebyshev polynomials and higher order Lucas Lehmer algorithm

Abstract

We extend the necessity part of Lucas Lehmer iteration for testing Mersenne prime to all base and uniformly for both generalized Mersenne and Wagstaff numbers(the later correspond to negative base). The role of the quadratic iteration $x \rightarrow x^2-2$ is extended by Chebyshev polynomial $T_n(x)$ with an implied iteration algorithm because of the compositional identity $T_n(T_m(x))=T_{nm}(x)$. This results from a Chebyshev polynomial primality test based essentially on the Lucas pair $(\omega_a,\overline{\omega}_a)$, $\omega_a=a+\sqrt{a^2-1}$, where $a \neq 0 \pm 1$. It seems interesting that the arithmetic are all coded in the Chebyshev polynomials $T_n(x)$.