A Proth number that is prime, i.e., a number of the form for odd ,
a positive integer, and .
Factors of Fermat numbers are of this form as long
as they satisfy the condition odd and . For example, the factor of is not a Proth prime since . (Otherwise, every odd prime would be a Proth prime.)
Proth primes satisfy Proth's theorem, i.e., a number of this form is prime iff
there exists a number a such that is congruent to modulo . This provides an easy computational test for Proth primes.
Yves Gallot has written a downloadable program for testing Proth primes and many
of the largest currently known primes have been found with this program.