A125739 Primes p such that 3^p + 3^((p + 1)/2) + 1 is prime.

3, 5, 7, 17, 19, 79, 163, 317, 353, 1049, 1759, 5153, 7541, 23743, 2237561, 4043119

Offset: 1

Author

Alexander Adamchuk, Dec 02 2006

Keywords

• hard
• more
• nonn

Comments

PrimePi[ a(n) ] = {2, 3, 4, 7, 8, 22, 38, 66, 71, 176, 274, 687, 956, ...}, the indices of the primes p.

a(17) > 4400000. - Serge Batalov, Jun 20 2023

Links

• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• S. S. Wagstaff, Jr., The Cunningham Project.

Crossrefs

Cf. A125738 = Primes p such that 3^p - 3^((p + 1)/2) + 1 is prime.

Cf. A007670 = Numbers n such that 2^n - 2^((n + 1)/2) + 1 is prime.

Cf. A007671 = Numbers n such that 2^n + 2^((n + 1)/2) + 1 is prime.

Cf. A066408 = Numbers n such that the Eisenstein integer has prime norm.

Programs

Magma

[p: p in PrimesUpTo(5000) | IsPrime(3^p+3^((p+1)div 2)+1)]; // Vincenzo Librandi, Oct 13 2014

Mathematica

Do[p=Prime[n];f=3^p+3^((p+1)/2)+1;If[PrimeQ[f],Print[{n,p}]],{n,1,200}]

PARI

lista(nn) = {forprime(p=3, nn, if (ispseudoprime(3^p + 3^((p + 1)/2) + 1), print1(p, ", ")););} \\ Michel Marcus, Oct 13 2014

Extensions

a(11)-a(13) from Stefan Steinerberger, Sep 08 2007

a(14) from Lelio R Paula (lelio(AT)sknet.com.br), May 07 2008

a(15) from Serge Batalov, Oct 12 2014

a(16) from Ryan Propper and Serge Batalov, Jun 20 2023