Lucasian criteria for the primality of $N=h\cdot 2^{n} -1$

Abstract:

Let $vi = v_{i - 1}^2 - 2$ with ${v_0}$ given. If ${v_{n - 2}} \equiv 0(\bmod N)$ is a necessary and sufficient criterion that $N = h \cdot {2^n} - 1$ be prime, this is called a Lucasian criterion for the primality of $N$. Many such criteria are known, but the case $h = 3A$ has not been treated in full generality earlier. A theorem is proved that (by aid of computer) enables the effective determination of suitable numbers ${v_0}$ for any given $N$, if $h < {2^n}$. The method is used on all $N$ in the domain $h = 3(6)105,n \leqq 1000$. The Lucasian criteria thus constructed are applied, and all primes $N = h \cdot {2^n} - 1$ in the domain are tabulated.