A Proth prime is a Proth number that is prime, i.e., a positive integer of the form 
for an odd positive
integer 
,
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 an integer 
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 an odd positive
integer 
such that 
is composite for every positive
integer 
.
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."