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Primes of the form x^2 + 32*x*y + y^2 for x and y nonnegative.
3

%I #12 Jun 10 2020 10:11:14

%S 229,349,409,421,661,769,829,1021,1069,1249,1381,1429,1549,1789,1801,

%T 1861,2089,2161,2269,2389,3001,3061,3109,3181,3229,3469,3889,4021,

%U 4129,4201,4441,4861,4909,5101,5449,5521,5869,5881,6121,6469,6481,6529,6781

%N Primes of the form x^2 + 32*x*y + y^2 for x and y nonnegative.

%C Are all terms == 1 mod 12? - _Zak Seidov_, Apr 25 2008

%C Yes: (i) all terms == 1 mod 3 because the quadratic form has terms == {0,1} mod 3 and the values ==0 mod 3 are not primes. (ii) all terms == 1 mod 4 because the quadratic form has terms == {0,1,2} mod 4 and the values = {0,2} mod 4 are not primes. By the Chinese remainder constructions for coprime 3 and 4 all prime terms are == 1 mod 12. - _R. J. Mathar_, Jun 10 2020

%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), discriminant 1020.

%t a = {}; w = 32; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

%Y Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

%K nonn

%O 1,1

%A _Artur Jasinski_, Apr 24 2008