|
%I #56 Feb 18 2022 16:01:52
%S 73,97,193,241,313,337,409,433,457,577,601,673,769,937,1009,1033,1129,
%T 1153,1201,1249,1297,1321,1489,1609,1657,1753,1777,1801,1873,1993,
%U 2017,2089,2113,2137,2161,2281,2377,2473,2521,2593,2617,2689,2713
%N Primes of the form x^2 + 24*y^2.
%C 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.
%C Discriminant = -96.
%C Also primes of the forms x^2 + 48*y^2 and x^2 + 72*y^2. See A140633. - _T. D. Noe_, May 19 2008
%C 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
%C 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
%C 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
%H Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, <a href="/A107008/b107008.txt">Table of n, a(n) for n = 1..10000</a> [First 1000 terms from Vincenzo Librandi, next 143 terms from N. J. A. Sloane]
%H P. L. Clark, J. Hicks, H. Parshall, K. Thompson, <a href="https://www.emis.de/journals/INTEGERS/papers/n37/n37.Abstract.html">GONI: primes represented by binary quadratic forms</a>, INTEGERS 13 (2013) #A37
%H D. A. Cox, <a href="http://www.math.toronto.edu/~ila/Cox-Primes_of_the_form_x2+ny2.pdf">Primes of the form x^2 + n*y^2</a>, A Wiley-Interscience publication, 1989
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H J. Voight, <a href="https://doi.org/10.1090/S0025-5718-07-01976-X">Quadratic forms that represent almost the same primes</a>, Math. Comp. 76 (2007) 1589-1617
%t QuadPrimes[1, 0, 24, 10000] (* see A106856 *)
%o (PARI) is(n) = isprime(n) && #qfbsolve(Qfb(1, 0, 24), n) == 2 \\ _David A. Corneth_, Jun 21 2020
%Y Subset of A033199 (2y here = y there).
%Y Is this the same as A141375?
%Y Cf. A000926, A002476, A007519, A111174, A139642.
%Y See also the cross-references in A140633.
%K nonn,easy
%O 1,1
%A _T. D. Noe_, May 09 2005
%E Recomputed b-file, deleted incorrect Mma program. - _N. J. A. Sloane_, Jun 08 2014
|
