Proth number
In number theory, a Proth number, named after the mathematician François Proth, is a number of the form
where 
is an odd positive integer and 
is a positive integer such that 
. Without the latter condition, all odd integers greater than 1 would be Proth numbers.
The first Proth numbers are (sequence A080075 in OEIS):
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):
The primality of a Proth number can be tested with Proth's theorem which states that a Proth number 
is prime if and only if there exists an integer 
for which the following is true:
The largest known Proth prime as of 2016 is 
, and is 3,918,990 digits long. It was found by Konstantin Agafonov in the Seventeen or Bust distributed computing project which announced it on 5 May 2007. It is also the largest known non-Mersenne prime.
See also
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