OFFSET
0,1
COMMENTS
Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2.
a(12) <= 4253, a(13) <= 7487, a(14) <= 71. - Daniel Suteu, Feb 07 2021
a(13) > 2500 and a(14) = 71. - Jinyuan Wang, Feb 07 2021
LINKS
EXAMPLE
No term is smaller than 3 (since 2 is the only smaller prime, and (2^(2^n) + 1)/2 is not an integer).
(3^(2^0) + 1)/2 = (3^1 + 1)/2 = (3 + 1)/2 = 4/2 = 2 is prime, so a(0)=3.
(3^(2^1) + 1)/2 = (3^2 + 1)/2 = 5 is prime, so a(1)=3.
(3^(2^2) + 1)/2 = (3^4 + 1)/2 = 41 is prime, so a(2)=3.
(3^(2^3) + 1)/2 = (3^8 + 1)/2 = 3281 = 17*193 is not prime, nor is (p^8 + 1)/2 for any other prime < 13, but (13^8 + 1)/2 = 407865361 is prime, so a(3)=13.
PROG
(PARI) a(n) = my(p=3); while (!isprime((p^(2^n) + 1)/2), p=nextprime(p+1)); p; \\ Michel Marcus, Feb 07 2021
(Alpertron) x=3; x=N(x); NOT IsPrime((x^8192+1)/2); N(x)
# Martin Ehrenstein, Feb 08 2021
(Python)
from sympy import isprime, nextprime
def a(n):
p, pow2 = 3, 2**n
while True:
if isprime((p**pow2 + 1)//2): return p
p = nextprime(p)
print([a(n) for n in range(9)]) # Michael S. Branicky, Mar 03 2021
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Jon E. Schoenfield, Feb 06 2021
EXTENSIONS
a(11) from Daniel Suteu, Feb 07 2021
a(12) from Jinyuan Wang, Feb 07 2021
a(13)-a(14), using Dario Alpern's integer factorization calculator and prior bounds, from Martin Ehrenstein, Feb 08 2021
STATUS
approved