Proth Prime
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 
.
 | OEIS | values of  for which  is
prime |
| 1 | 1, 2, 4, 8, 16, ... | |
| 3 | A002253 | 1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, ... |
| 5 | A002254 | 1, 3, 7, 13, 15, 25, 39, 55, 75, 85, 127, 1947, ... |
| 7 | A032353 | 2, 4, 6, 14, 20, 26, 50, 52, 92, 120, ... |
| 9 | A002256 | 1, 2, 3, 6, 7, 11, 14, 17, 33, 42, 43, 63, ... |
Ballinger, R. "Proth Search Page." http://www.prothsearch.net/.
Caldwell, C. "Proth Prime." http://primes.utm.edu/glossary/page.php?sort=ProthPrime.
Caldwell, C. K. "Yves Gallot's Proth.exe: An implementation of Proth's Theorem for Windows." http://www.utm.edu/research/primes/programs/gallot/.
Keller, W. "The Least Prime of the Form 
for Certain
Values of 
." Abstr. Amer. Math. Soc. 9,
417-418, 1988.
McNamara, J. and Mills, M. "Factoring of Proth Numbers." http://www.fidn.org/proth1.html.
Sloane, N. J. A. Sequences A002253/M1318, A002254/M2635, A002256/M0751, A032353, and A080076 in "The On-Line Encyclopedia of Integer Sequences."
Weisstein, Eric W. "Proth Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProthPrime.html
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Apéry's constant