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A141215
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Primes of the form 3*x^2+5*x*y-3*y^2 (as well as 5*x^2+9*x*y+y^2).
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6
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3, 5, 13, 19, 41, 47, 61, 73, 83, 97, 103, 107, 109, 113, 127, 131, 137, 149, 163, 167, 179, 197, 199, 229, 239, 241, 257, 263, 269, 271, 283, 293, 317, 347, 353, 367, 379, 431, 439, 443, 449, 461, 463, 479, 487, 491, 503, 563, 569, 571, 601, 607, 613, 619
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OFFSET
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1,1
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COMMENTS
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Discriminant = 61. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
3*x^2+5*x*y-3*y^2 and 5*x^2+9*x*y+y^2 are equivalent forms.
Also, primes of the form x^2 - 61y^2, of discriminant 244.
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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(8) = 73 because we can write 73 = 3*4^2+5*4*5-3*5^2 (or 73 = 5*3^2+9*3*1+1^2).
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MAPLE
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select(p -> isprime(p) and nops([isolve(x^2 - 61*y^2 = p)])>0, [seq(2*i+1, i=1..1000)]); # Robert Israel, Jun 11 2014
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MATHEMATICA
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terms = 100; d = 61;
Table[3*x^2 + 5*x*y - 3*y^2, {x, 1, terms}, {y, Floor[(5 - Sqrt[d])*x/6], Ceiling[(5 + Sqrt[d])*x/6]}] // Flatten // Select[#, Positive[#] && PrimeQ[#]&]& // Union // Take[#, terms]& (* Jean-François Alcover, Feb 28 2019 *)
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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 (sergarmor(AT)yahoo.es), Jun 14 2008
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STATUS
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approved
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