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# Proth number

In number theory, a Proth number is a number of the form

${\displaystyle N=k\cdot 2^{n}+1}$

where ${\displaystyle k}$ is an odd positive integer and ${\displaystyle n}$ is a positive integer such that ${\displaystyle 2^{n}>k}$. They are named after the mathematician François Proth. The first few Proth numbers are

3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241 (sequence A080075 in the OEIS).

The Cullen numbers (numbers of the form n·2n + 1) and Fermat numbers (numbers of the form 22n + 1) are special cases of Proth numbers. Without the condition that ${\displaystyle 2^{n}>k}$, all odd integers greater than 1 would be Proth numbers.[1]

## Proth primes

A Proth prime is a Proth number which is prime. The first few Proth primes are

3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 ().

The primality of a Proth number can be tested with Proth's theorem, which states[2] that a Proth number ${\displaystyle p}$ is prime if and only if there exists an integer ${\displaystyle a}$ for which

${\displaystyle a^{\frac {p-1}{2}}\equiv -1{\pmod {p}}.}$

The largest known Proth prime as of 2016 is ${\displaystyle 10223\cdot 2^{31172165}+1}$, and is 9,383,761 digits long.[3] It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016.[4] It is also the largest known non-Mersenne prime.[5]