A127924 One-seventh of the difference of squares of legs of primitive Pythagorean triangles, neither of which is a multiple of 7.

1, 17, 23, 103, 137, 199, 217, 497, 601, 697, 799, 1343, 1457, 1679, 1799, 2737, 2839, 2921, 3199, 3337, 3503, 3503, 3937, 3961, 3977, 4183, 4577, 4657, 5543, 6103, 6463, 7399, 7663, 8143, 8977, 9143, 9881, 10097, 10577, 10897, 10943, 11543, 13703, 13817

Offset

1,2

Comments

If 7 divides neither m nor n, then from Fermat's little theorem, 7 divides M^3 - N^3 =( alpha)*Q, where alpha= M - N and Q=M^2 + M*N + N^2, with M=m^2, N=n^2; Here we have (alpha)^2 + (beta)^2 = (gamma)^2, with (beta)^2 =4*M*N and (gamma)^2=M + N. Thus if further 7 does not divide alpha, then 7 divides Q - 7M*N=(M - N)^2 - 4*M*N=(alpha)^2 - (beta)^2, so that 7 always divides (alpha)*(beta)*(alpha^2 - beta^2). In a primitive Pythagorean triangle, 7 divides one of the legs or their sum or their difference.

3503 appears twice in the sequence for (a,b)=(52,165) and (1748,1755). - Chandler

Links

Table of n, a(n) for n=1..44.

Crossrefs

Cf. A127923.

Sequence in context: A260179 A100473 A060252 * A250640 A344636 A108260

Adjacent sequences: A127921 A127922 A127923 * A127925 A127926 A127927

Keyword

nonn

Author

Lekraj Beedassy, Feb 06 2007

Extensions

Extended by Ray Chandler, Apr 11 2010

Status

approved