Ward's Primality Test

Let $N$ be an Odd Integer, and assume there exists a Lucas Sequence $\{U_n\}$ with associated Sylvester Cyclotomic Numbers $\{Q_n\}$ such that there is an $n>\sqrt{N}$ (with $n$ and $N$ Relatively Prime) for which $N$ Divides $Q_n$. Then $N$ is a Prime unless it has one of the following two forms:

1. $N=(n-1)^2$, with $n-1$ Prime and $n>4$, or

2. $N=n^2-1$, with $n-1$ and $n+1$ Prime.

See also Lucas Sequence, Sylvester Cyclotomic Number

References

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 69-70, 1989.