Primality testing using cyclotomic polynomials

Mathematics Asked on August 2, 2020

Can you prove or disprove the following claim:

Let $Phi_m(x)$ be the mth cyclotomic polynomial , and let $n$ be a natural number greater than one . If there exists an integer $a$ such that $$Phi_{n-1}(a) equiv 0 pmod{n}$$ then $n$ is prime.

You can run this test here. I have verified this claim for all prime numbers less than $1000000$ .

One Answer

Quoting from Dickson, History of the Theory of Numbers, Volume 1, page 378:

A. Hurwitz [L'intermediaire des math. 2 (1895) 41] gave a generalization of Proth's theorem. Let $F_n(x)$ denote an irreducible factor of degree $phi(n)$ of $x^n-1$. Then if there exists an integer $q$ such that $F_{p-1}(q)$ is divisible by $p$, $p$ is a prime.

I haven't been able to find the Hurwitz paper.

Correct answer by Gerry Myerson on August 2, 2020

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