Proth prime

Proth prime
Named after	François Proth
Publication year	1878
Author of publication	Proth, Francois
No. of known terms	4304683178 below 272
Conjectured no. of terms	Infinite
Subsequence of	Proth numbers, prime numbers
Formula	k × 2n + 1
First terms	3, 5, 13, 17, 41, 97, 113
Largest known term	10223 × 231172165 + 1 (as of December 2019)
OEIS index	A080076; Proth primes: primes of the form k*2^m + 1 with odd k < 2^m, m ≥ 1;

A Proth number is a natural number N of the form $N=k\times 2^{n}+1$ where k and n are positive integers, k is odd and $2^{n}>k$ . A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth. 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 (OEIS: A080076).

It is still an open question whether an infinite number of Proth primes exist. It was shown in 2022 that the reciprocal sum of Proth primes converges to a real number near 0.747392479, substantially less than the value of 1.093322456 for the reciprocal sum of Proth numbers.

The primality of Proth numbers can be tested more easily than many other numbers of similar magnitude.

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