Proth number

In number theory, a Proth number is a number of the form In number theory, a Proth number is a number of the form where k {displaystyle k} is an odd positive integer and n {displaystyle n} is a positive integer such that 2 n > k {displaystyle 2^{n}>k} . They are named after the mathematician François Proth. The first few Proth numbers are 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 2 n > k {displaystyle 2^{n}>k} , all odd integers greater than 1 would be Proth numbers. A Proth prime is a Proth number which is prime. The first few Proth primes are The primality of a Proth number can be tested with Proth's theorem, which states that a Proth number p {displaystyle p} is prime if and only if there exists an integer a {displaystyle a} for which The largest known Proth prime as of 2016 is 10223 ⋅ 2 31172165 + 1 {displaystyle 10223cdot 2^{31172165}+1} , and is 9,383,761 digits long. It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016. It is also the largest known non-Mersenne prime.