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.

External links

• Wikipedia