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

1,1

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

Mathematica

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

Prog

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

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.

Sequence in context: A139972 A268426 A155573 * A141375 A140621 A143577

Adjacent sequences: A107005 A107006 A107007 * A107009 A107010 A107011

Keyword

nonn,easy

Author

T. D. Noe, May 09 2005

Extensions

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

Status

approved