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A106865
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Primes of the form 2x^2 + 2xy + 3y^2.
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13
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2, 3, 7, 23, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 263, 283, 307, 347, 367, 383, 443, 463, 467, 487, 503, 523, 547, 563, 587, 607, 643, 647, 683, 727, 743, 787, 823, 827, 863, 883, 887, 907, 947, 967, 983, 1063, 1087, 1103, 1123, 1163, 1187
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OFFSET
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1,1
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COMMENTS
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Discriminant = -20.
Also: Primes of the form 2x^2 - 2xy + 3y^2 with x and y nonnegative. Cf. A106864.
Primes congruent to 2, 3, 7 modulo 20. - Michael Somos, Aug 13 2006
In Z[sqrt(-5)], these numbers are irreducible but not prime. In terms of ideals, they generate principal ideals that are not prime (or maximal). The equation x^2 + 5y^2 = a(n) has no solutions, but x^2 = -5 (mod a(n)) does. For example, 2 * 3 = (1 - sqrt(-5))(1 + sqrt(-5)) and 7 * 23 = (9 - 4*sqrt(-5))(9 + 4*sqrt(-5)). - Alonso del Arte, Dec 19 2015
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REFERENCES
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David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.
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LINKS
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FORMULA
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EXAMPLE
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x = 1, y = 1 gives 2x^2 + 2xy + 3y^2 = 2 + 2 + 3 = 7.
x = 1, y = -3 gives 2x^2 + 2xy + 3y^2 = 2 - 6 + 27 = 23.
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MAPLE
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select(isprime, [2, seq(seq(5+s+20*i, s=[-2, 2]), i=0..10^3)]); # Robert Israel, Dec 23 2015
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MATHEMATICA
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QuadPrimes2[2, -2, 3, 10000] (* see A106856 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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