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.
A Sierpiński number of the second kind is a number 
satisfying Sierpiński's composite number theorem, i.e., a Proth number 
such that 
is composite for every 
.
The first few Proth primes are 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, ... (OEIS A080076).
The following table gives the first few indices 
such that 
is prime for small 
.
for which 
is prime
for Certain Values of 
." Abstr. Amer. Math. Soc. 9, 417-418, 1988.McNamara, J. and Mills, M. "Factoring of Proth Numbers."