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# Proth's theorem

This article is about **Proth's theorem**.

Proth's theorem (1878) states:

Let [math]\displaystyle{ n = h*2^k+1 }[/math] and [math]\displaystyle{ h\lt 2^k }[/math]; then [math]\displaystyle{ n }[/math] is prime if (and only if) there is an integer [math]\displaystyle{ a }[/math] such that

- [math]\displaystyle{ a^{(n-1)/2} \equiv -1 (mod\,n) }[/math].

A prime of this form is known as a Proth prime.