A139642 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) is a convenient number.

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

Offset: 1

Author

T. D. Noe, Apr 28 2008

Keywords

• fini
• nonn
• tabf

Comments

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).

Examples

			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,
...
		

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Section 3.

Links

• T. D. Noe, Rows n=1..65 of triangle, flattened
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

See the Binary Quadratic Forms and OEIS link for full list of primes generated by x^2+Ny^2, where N is a convenient number.