A139490 - OEIS

%I #18 Mar 13 2023 13:30:19

%S 1,4,6,7,8,10,14,16,18,22,26,38,58,82,86

%N Numbers n such that the quadratic form x^2 + n*x*y + y^2 represents exactly the same primes as the quadratic form x^2 + m*y^2 for some m.

%C For the numbers m see A139491.

%C Conjecture: This sequence is finite and complete (checked for range n<=200 and m<=500).

%C Three more terms were found by searching n <= 1000 and m <= 4000. The corresponding m are 840, 840, and 1848, which are idoneal numbers A000926. The sequence is probably complete now. [_T. D. Noe_, Apr 27 2009]

%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)

%e a(1)=1 because the primes represented by x^2+xy+y^2 are the same as the primes represented by x^2 + 3*y^2 (see A007645).

%e The known pairs (n,m) are the following (checked for range n<=200 and m<=500):

%e n={1, 4, 4, 6, 6, 7, 8, 8, 10, 10, 10, 14, 14, 14, 16, 18, 22, 22, 26, 38, 38}

%e m={3, 9, 12, 8, 16, 15, 45, 60, 24, 48, 72, 24, 48, 72, 21, 40, 120, 240, 168, 120, 240}.

%t f = 200; g = 300; h = 30; j = 100; b = {}; Do[a = {}; Do[Do[If[PrimeQ[x^2 + n y^2], AppendTo[a, x^2 + n y^2]], {x, 0, g}], {y, 1, g}]; AppendTo[b, Take[Union[a], h]], {n, 1, f}]; Print[b]; c = {}; Do[a = {}; Do[Do[If[PrimeQ[n^2 + w*n*m + m^2], AppendTo[a, n^2 + w*n*m + m^2]], {n, m, g}], {m, 1, g}]; AppendTo[c, Take[Union[a], h]], {w, 1, j}]; Print[c]; bb = {}; cc = {}; Do[Do[If[b[[p]] == c[[q]], AppendTo[bb, p]; AppendTo[cc, q]], {p, 1, f}], {q, 1, j}]; Union[cc] (*Artur Jasinski*)

%Y Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139491.

%K nonn

%O 1,2

%A _Artur Jasinski_, Apr 24 2008, Apr 26 2008, Apr 27 2008

%E Edited by _N. J. A. Sloane_, Apr 25 2008

%E Extended by _T. D. Noe_, Apr 27 2009

%E Typo fixed by _Charles R Greathouse IV_, Oct 28 2009