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Search: id:A139492
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| A139492 |
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Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative. |
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+0
4
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| 7, 37, 43, 67, 79, 109, 127, 151, 163, 193, 211, 277, 331, 337, 373, 379, 421, 457, 463, 487, 499, 541, 547, 571, 613, 631, 673, 709, 739, 751, 757, 823, 877, 883, 907, 919, 967, 991, 1009, 1033, 1051, 1087, 1093, 1117, 1129, 1171, 1201, 1213, 1297, 1303
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Values of the quadratic form are {0,1,3,4} mod 6, so this is a subset of A002476. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008
Is this the same sequence as A002476?
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REFERENCES
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D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
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MAPLE
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It can be checked using Maple that the primes p of the form x^2+n*x*y+y^2, n >=3, where x and y are nonnegative, depend on n mod 6 as follows: n mod 6 = 0 => p mod 12 = {1, 5}; n mod 6 = 1 => p mod 12 = {1, 7}; n mod 6 = 2 => p mod 12 = {1}; n mod 6 = 3 => p mod 12 = {1, 5, 7, 11}; n mod 6 = 4 => p mod 12 = {1}; n mod 6 = 5 => p mod 12 = {1, 7}. Keep in mind that 12 is a canonical base for mathematics in general since any prime greater than 3 is of the form 6k+-1, any prime of the form 4k+1 is a sum of squares while any prime of the form 4k+3 is never a sum of squares and lcm(6, 4)=12. - Walter A. Kehowski (wkehowski(AT)cox.net), Jun 01 2008
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MATHEMATICA
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a = {}; w = 5; 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]
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CROSSREFS
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Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.
Sequence in context: A038478 A043010 A141159 this_sequence A092475 A106924 A076285
Adjacent sequences: A139489 A139490 A139491 this_sequence A139493 A139494 A139495
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KEYWORD
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nonn
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AUTHOR
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Artur Jasinski (grafix(AT)csl.pl), Apr 24 2008
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