Proth number - Alchetron, The Free Social Encyclopedia

In number theory, a Proth number, named after the mathematician François Proth, is a number of the form

N = k ⋅ 2 n + 1

where k is an odd positive integer and n is a positive integer such that 2 n > k . Without the latter condition, all odd integers greater than 1 would be Proth numbers.

The first few Proth numbers are (sequence A080075 in the OEIS):

3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, etc.

The Cullen numbers (n·2ⁿ+1) and Fermat numbers (2^2ⁿ+1) are special cases of Proth numbers.

Proth primes

A Proth prime is a Proth number which is prime. The first few Proth primes are ( A080076):

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 that a Proth number p is prime if and only if there exists an integer a for which the following is true:

a p − 1 2 ≡ − 1 ( mod p ) .

The largest known Proth prime as of 2016 is 10223 ⋅ 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.

References

Proth number Wikipedia

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