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A107008
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Primes of the form x^2 + 24*y^2.
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36
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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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]
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MATHEMATICA
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QuadPrimes[1, 0, 24, 10000] (* see A106856 *)
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PROG
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(PARI) is(n) = isprime(n) && #qfbsolve(Qfb(1, 0, 24), n) == 2 \\ David A. Corneth, Jun 21 2020
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CROSSREFS
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Subset of A033199 (2y here = y there).
See also the cross-references in A140633.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Recomputed b-file, deleted incorrect Mma program. - N. J. A. Sloane, Jun 08 2014
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STATUS
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approved
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