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%I #26 Sep 08 2022 08:45:33
%S 103,127,151,223,271,409,433,457,463,577,631,727,769,919,937,967,1033,
%T 1039,1063,1087,1249,1279,1327,1447,1471,1543,1657,1753,1759,1777,
%U 1783,1801,1879,1951,1993,2089,2143,2161,2287,2311,2473,2503,2551
%N Primes of the form x^2+Ny^2, with N=102.
%C Discriminant=-408. N is an idoneal number (A000926), which means that the quadratic form's genus consists of a single class, which means that the primes of this form are identical to the primes that are congruent to c (mod 4N), where c is a set of numbers less than 4N. The sequence A139642 lists the set c for each idoneal number. That sequence also cross references the sequences for the quadratic forms with N equal to the first 36 idoneal numbers. The remaining quadratic forms are this sequence and the 28 listed in order below. Note that the sequences for N=120 and 240 are the same.
%C The primes are congruent to {1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361} (mod 408).
%D David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
%D L. E. Dickson, History of the Theory of Numbers, Vol 3, Chelsea, 1923.
%H Vincenzo Librandi and Ray Chandler, <a href="/A139643/b139643.txt">Table of n, a(n) for n = 1..10000</a> [First 1000 terms from Vincenzo Librandi]
%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)
%p C:= [1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361]:
%p select(isprime, [seq(seq(408*i+j,j=C),i=0..100)]); # _Robert Israel_, Jul 03 2016
%t nn=102; pMax=10000; Union[Reap[Do[p=x^2+nn*y^2; If[p<=pMax && PrimeQ[p], Sow[p]], {x,Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2,1]]] (* _T. D. Noe_, Aug 02 2009 *)
%t QuadPrimes2[1, 0, 102, 10000] (* see A106856 *)
%o (Magma) [ p: p in PrimesUpTo(3000) | p mod 408 in {1, 25, 49, 55, 103, 121, 127, 145, 151, 169, 217, 223, 247, 271, 319, 361}]; // _Vincenzo Librandi_, Jul 28 2012
%o (Magma) k:=102; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // _Bruno Berselli_, Jun 01 2016
%Y Cf. A139644, A139645, A139502, A139646, A139647, A139648, A139506, A139649, A139650, A139651, A139652, A139502, A139653-A139668.
%K nonn,easy
%O 1,1
%A _T. D. Noe_, Apr 29 2008
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