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A141183 Primes of the form -x^2+6*x*y+2*y^2 (as well as of the form 7*x^2+10*x*y+2*y^2). 8

%I #32 Feb 17 2022 11:36:41

%S 2,7,11,19,43,79,83,107,127,131,139,151,167,211,227,239,263,271,283,

%T 307,347,359,431,439,479,491,503,523,547,563,571,607,659,739,743,787,

%U 811,827,887,919,967,1019,1031,1051,1063,1091,1151,1163,1187,1223,1231,1283,1319,1327

%N Primes of the form -x^2+6*x*y+2*y^2 (as well as of the form 7*x^2+10*x*y+2*y^2).

%C Discriminant = 44. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

%C Also primes of form 11*u^2-v^2. The transformation {u,v}={-3*x-y,10*x+3*y} yields the form in the title. - _Juan Arias-de-Reyna_, Mar 20 2011

%C Also primes p equal -1 mod 4 and = 1, 3, 4, 5, or 9 mod 11. - _Juan Arias-de-Reyna_, Mar 20 2011

%D Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, NY, 1966.

%H Juan Arias-de-Reyna, <a href="/A141183/b141183.txt">Table of n, a(n) for n = 1..10000</a>

%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. OEIS wiki, June 2014.

%H D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.

%e a(4)=19 because we can write 19= -1^2+6*1*2+2*2^2 (or 19=7*1^2+10*1*1+2*1^2).

%t Select[Prime[Range[250]], # == 2 || # == 11 || MatchQ[Mod[#, 44], Alternatives[7, 19, 35, 39, 43]]&] (* _Jean-François Alcover_, Oct 28 2016 *)

%Y Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141182 (d=44).

%Y For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

%K nonn

%O 1,1

%A Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 13 2008

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Last modified March 28 11:46 EDT 2024. Contains 371241 sequences. (Running on oeis4.)