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

3, 11, 193, 239, 659, 709, 1103, 2029, 9049, 10453, 255361, 534827, 2888387

Offset: 1

Author

Alexander Adamchuk, Dec 02 2006

Keywords

• hard
• more
• nonn

Comments

PrimePi[ a(n) ] = {2, 5, 44, 52, 120, 127, 185, 308, 1125, 1278 ...}, the indices of the primes p.

a(14) > 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. A125739 = 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

Mathematica

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

Extensions

More terms from A066408 by Serge Batalov, Mar 24 2014

a(13) from Ryan Propper and Serge Batalov, Jun 20 2023