A107008 Primes of the form x^2 + 24*y^2.

73, 97, 193, 241, 313, 337, 409, 433, 457, 577, 601, 673, 769, 937, 1009, 1033, 1129, 1153, 1201, 1249, 1297, 1321, 1489, 1609, 1657, 1753, 1777, 1801, 1873, 1993, 2017, 2089, 2113, 2137, 2161, 2281, 2377, 2473, 2521, 2593, 2617, 2689, 2713

Offset: 1

Author

T. D. Noe, May 09 2005

Keywords

• nonn
• easy

Comments

Presumably this is the same as primes congruent to 1 mod 24, so a(n) = 24*A111174(n) + 1. - N. J. A. Sloane, Jul 11 2008. Checked for all terms up to 2 million. - Vladimir Joseph Stephan Orlovsky, May 18 2011.

Discriminant = -96.

Also primes of the forms x^2 + 48*y^2 and x^2 + 72*y^2. See A140633. - T. D. Noe, May 19 2008

Primes of the quadratic form are a subset of the primes congruent to 1 (mod 24). [Proof. For 0 <= x, y <= 23, the only values mod 24 that x^2 + 24*y^2 can take are 0, 1, 4, 9, 12 or 16. All of these r except 1 have gcd(r, 24) > 1 so if x^2 + 24*y^2 is prime its remainder mod 24 must be 1.] - David A. Corneth, Jun 08 2020

More advanced mathematics seems to be needed to determine whether this sequence lists all primes congruent to 1 (mod 24). Note the significance of 24 being a convenient number, as described in A000926. See also Sloane et al., Binary Quadratic Forms and OEIS, which explains how the table in A139642 may be used for this determination. - Peter Munn, Jun 21 2020

Primes == 1 (mod 2^3*3) are the intersection of the primes == 1 (mod 2^3) in A007519 and the primes == 1 (mod 3) in A002476, by the Chinese remainder theorem. - R. J. Mathar, Jun 11 2020

Links

• Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 143 terms from N. J. A. Sloane]
• P. L. Clark, J. Hicks, H. Parshall, K. Thompson, GONI: primes represented by binary quadratic forms, INTEGERS 13 (2013) #A37
• D. A. Cox, Primes of the form x^2 + n*y^2, A Wiley-Interscience publication, 1989
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• J. Voight, Quadratic forms that represent almost the same primes, Math. Comp. 76 (2007) 1589-1617

Crossrefs

Subset of A033199 (2y here = y there).

Is this the same as A141375?

Cf. A000926, A002476, A007519, A111174, A139642.

See also the cross-references in A140633.

Programs

Mathematica

QuadPrimes[1, 0, 24, 10000] (* see A106856 *)

PARI

is(n) = isprime(n) && #qfbsolve(Qfb(1, 0, 24), n) == 2 \\ David A. Corneth, Jun 21 2020

Extensions

Recomputed b-file, deleted incorrect Mma program. - N. J. A. Sloane, Jun 08 2014