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Proth's theorem
This article is about Proth's theorem.
Proth's theorem (1878) states:
Let [math]\displaystyle{ p = k*2^n+1 }[/math] and [math]\displaystyle{ k \lt 2^n }[/math]; then [math]\displaystyle{ p }[/math] is prime if there is an integer [math]\displaystyle{ a }[/math] such that
- [math]\displaystyle{ a^{(p-1)/2} \equiv -1\pmod{p} }[/math].
Furthermore, if [math]\displaystyle{ a }[/math] is a quadratic non-residue modulo [math]\displaystyle{ p }[/math], then the converse is also true.
A prime [math]\displaystyle{ p }[/math] of this form is known as a Proth prime.