Proth number

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

$N=k \cdot 2^n+1$

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 Proth numbers are (sequence A080075 in 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 (2²ⁿ+1) are special cases of Proth numbers.

Proth primes

A Proth prime is a Proth number which is prime. The first 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:

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

The largest known Proth prime as of 2010 is $19249 \cdot 2^{13018586} + 1$. It was found by Konstantin Agafonov in the Seventeen or Bust distributed computing project which announced it 5 May 2007. It is also the largest known non-Mersenne prime.

See also

• Sierpinski number
• PrimeGrid – a distributed computing project searching for large Proth primes