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Search: id:A139642
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| A139642 |
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Irregular triangle where row n gives the congruence (mod 4N) for the primes represented by the quadratic form x^2+Ny^2, where N=A000926(n). |
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+0
4
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| 1, 2, 1, 2, 3, 1, 3, 7, 1, 5, 9, 13, 1, 5, 9, 1, 7, 1, 7, 9, 11, 15, 23, 25, 1, 9, 17, 25, 1, 13, 25, 1, 9, 11, 19, 1, 13, 25, 37, 1, 9, 13, 17, 25, 29, 49, 1, 19, 31, 49, 1, 9, 17, 25, 33, 41, 49, 57, 1, 19, 25, 43, 49, 67, 1, 25, 37, 1, 9, 15, 23, 25, 31, 47, 49, 71, 81, 1, 25, 49, 73, 1, 9
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Each row begins with 1. For example, the 12th row is for N=13. The numbers in that row are 1, 9, 17, 25, 29,and 49, which means that the primes represented by the quadratic form x^2+13y^2 (A033210) are congruent to 1, 9, 17, 25, 29,or 49 (mod 52). Cox lists some of these congruences on page 36 of his book. As mentioned by Cox, for these N, every term of the congruence has the form b^2 or N+b^2 for some integer b. In some cases, the congruences can be simplified. For instance, for N=18 (A106950), the congruence is 1, 19, 25, 43, 49, 67 (mod 72), which can be simplified to 1, 19 (mod 24).
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REFERENCES
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David A. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989, Section 3.
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LINKS
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T. D. Noe, Rows n=1..65 of triangle, flattened
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EXAMPLE
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1,2; 1,2,3; 1,3,7; 1,5,9,13; 1,5,9; 1,7; 1,7,9,11,15,23,25; 1,9,17,25
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CROSSREFS
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Cf. A002313, A033200, A007645, A002144, A033205, A033199, A033207, A007519, A068228, A033201, A068228, A033210, A033212, A007519, A106950, A033215, A033216, A107008, A033205, A107134, A033220, A033222, A033225, A107145, A033229, A107152, A107008, A033236, A033237, A107152, A033245, A107008, A033250, A033254, A107202, A033202 (the quadratic forms x^2+Ny^2 for N up to 93).
Cf. A139643-A139668 (idoneal quadratic forms for N > 93).
Sequence in context: A104468 A110062 A122087 this_sequence A021475 A132224 A133299
Adjacent sequences: A139639 A139640 A139641 this_sequence A139643 A139644 A139645
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KEYWORD
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fini,nonn,tabf
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Apr 28 2008
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