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Difference between revisions of "Proth's theorem"
(typos fixed) |
(Changing notation to modern form and correcting converse) |
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Proth's theorem (1878) states: | Proth's theorem (1878) states: | ||
| − | Let <math> | + | Let <math>p = k*2^n+1</math> and <math>k < 2^n</math>; then <math>p</math> is prime if there is an integer <math>a</math> such that |
| − | :<math>a^{( | + | :<math>a^{(p-1)/2} \equiv -1\pmod{p}</math>. |
| − | A prime of this form is known as a [[Proth prime]]. | + | Furthermore, if <math>a</math> is a [[quadratic residue|quadratic non-residue]] modulo <math>p</math>, then the converse is also true. |
| + | |||
| + | A prime <math>p</math> of this form is known as a [[Proth prime]]. | ||
==External links== | ==External links== | ||
*[[Wikipedia:Proth's theorem|Wikipedia]] | *[[Wikipedia:Proth's theorem|Wikipedia]] | ||
| + | |||
[[Category:Primality tests]] | [[Category:Primality tests]] | ||
Revision as of 18:15, 28 September 2023
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.
