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A141180
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Primes of the form x^2+6*x*y-y^2 (as well as of the form 6*x^2+8*x*y+y^2).
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7
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31, 41, 71, 79, 89, 151, 191, 199, 239, 241, 271, 281, 311, 359, 401, 409, 431, 439, 449, 479, 521, 569, 599, 601, 631, 641, 719, 751, 761, 769, 809, 839, 881, 911, 919, 929, 991, 1009, 1031, 1039, 1049, 1129, 1151, 1201, 1231, 1249, 1279, 1289, 1319, 1321, 1361, 1399, 1409, 1439
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OFFSET
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1,1
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COMMENTS
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Discriminant = 40. 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.
Also primes of form 10*u^2 - v^2. The transformation {u, v} = {-x, 3*x-y} yields the form in the title, and primes of form U^2 - 10*V^2, with transformation {U, V} = {x+3*y, y}. - Juan Arias-de-Reyna, Mar 19 2011
Therefore, these primes are composite in Q(sqrt(10)), as they can be factored thus: (-u + v*sqrt(10))*(u + v*sqrt(10)). - Alonso del Arte, Jul 22 2012
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REFERENCES
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Z. I. Borevich and I. R. Shafarevich, Number Theory.
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LINKS
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EXAMPLE
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a(2) = 41 because we can write 41 = 3^2 + 6*3*2 - 2^2 (or 41 = 6*2^2 + 8*2*1 + 1^2). Furthermore, notice that (-7 + 3*sqrt(10))(7 + 3*sqrt(10)) = 41.
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MATHEMATICA
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Take[Select[Union[Flatten[Table[Abs[a^2 - 10b^2], {a, 0, 49}, {b, 0, 49}]]], PrimeQ], 50] (* Alonso del Arte, Jul 22 2012 *)
Select[Prime[Range[250]], MatchQ[Mod[#, 40], Alternatives[1, 9, 31, 39]]&] (* Jean-François Alcover, Oct 28 2016 *)
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CROSSREFS
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For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
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KEYWORD
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nonn
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AUTHOR
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Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (lourdescm84(AT)hotmail.com), Jun 12 2008
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EXTENSIONS
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STATUS
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approved
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