A089682 Primes of the form 3*m^2 - 1.

2, 11, 47, 107, 191, 431, 587, 971, 1451, 2027, 2351, 2699, 3467, 4799, 5807, 6911, 7499, 8111, 8747, 10091, 10799, 14699, 15551, 16427, 17327, 18251, 25391, 27647, 36299, 41771, 44651, 55487, 57131, 62207, 67499, 71147, 74891, 80687, 92927, 99371, 103787, 106031

Offset: 1

Author

Giovanni Teofilatto, Jan 05 2004

Keywords

• nonn
• easy

Comments

431 and 27647 also have the form 2*m^3-1 (431 = 3*12^2-1 = 2*6^3-1 and 27647 = 3*96^2-1 = 2*24^3-1). [Howard Berman (howard_berman(AT)hotmail.com), May 09 2009]

Or, primes p such that 3*(p+1) is a square. [Vincenzo Librandi, Nov 18 2010]

Obviously m must be either 1 or an even number. So this consists of 1 together with primes of the form 12*u^2 - 1. - N. J. A. Sloane, Aug 09 2019

References

• M. Cerasoli, F. Eugeni, and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
• Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Primes in A080663.

Cf. A089681.

Programs

Magma

[a: n in [0..200] | IsPrime(a) where a is 3*n^2-1]; // Vincenzo Librandi, Jul 16 2012

Mathematica

Select[Table[3 n^2 - 1, {n, 0, 2000}], PrimeQ] (* Vincenzo Librandi, Jul 16 2012 *)

Formula

a(n) = A080663(A089681(n)). - Elmo R. Oliveira, Feb 22 2025

Extensions

More terms from Rick L. Shepherd, May 18 2005