A275530 Smallest positive integer m such that (m^(2^n) + 1)/2 is prime.

3, 3, 3, 9, 3, 3, 3, 113, 331, 513, 827, 799, 3291, 5041, 71, 220221, 23891, 11559, 187503, 35963

Offset: 0

Author

Walter Kehowski, Jul 31 2016

Keywords

• nonn
• more

Comments

The terms of this sequence with n > 11 correspond to probable primes which are too large to be proven prime currently. - Serge Batalov, Apr 01 2018

a(15) is a statistically significant outlier; the sequence (m^(2^15)+1)/2 may require a double-check with software that is not GWNUM-based. - Serge Batalov, Apr 01 2018

Examples

			a(7) = 113 since 113 is the smallest positive integer m such that (m^(2^7)+1)/2 is prime.
		

Links

• Richard Fischer, Generalized Fermat numbers with odd base
• Wikipedia, Fermat number

Crossrefs

Cf. A056993, A027862.

Programs

Maple

a:= proc(n) option remember; local m; for m by 2
      while not isprime((m^(2^n)+1)/2) do od; m
    end:
seq(a(n), n=0..8);

Mathematica

Table[m = 1; While[! PrimeQ[(m^(2^n) + 1)/2], m++]; m, {n, 0, 9}] (* Michael De Vlieger, Sep 23 2016 *)

PARI

a(n) = {my(m = 1); while (! isprime((m^(2^n)+1)/2), m += 2); m;} \\ Michel Marcus, Aug 01 2016

Python

from sympy import isprime
def a(n):
  m, pow2 = 1, 2**n
  while True:
    if isprime((m**pow2 + 1)//2): return m
    m += 2
print([a(n) for n in range(9)]) # Michael S. Branicky, Mar 03 2021

Extensions

a(13)-a(14) from Robert Price, Sep 23 2016

a(15) from Serge Batalov, Mar 29 2018

a(16) from Serge Batalov, Mar 30 2018

a(17) from Serge Batalov, Apr 01 2018

a(18)-a(19) from Ryan Propper, Aug 16 2022. These correspond to 1382288- and 2388581-digit PRPs, respectively, found using an exhaustive search with Jean Penne's LLR2.