Proth prime
Prime number of the form k*(2^n)+1 / From Wikipedia, the free encyclopedia
A Proth number is a natural number N of the form where k and n are positive integers, k is odd and . A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth.[2] 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).
Quick Facts Named after, Publication year ...
Named after | François Proth |
---|---|
Publication year | 1878 |
Author of publication | Proth, Francois |
No. of known terms | 4304683178 below 272 [1] |
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 |
|
Close
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.[1]
The primality of Proth numbers can be tested more easily than many other numbers of similar magnitude.