A127923 -id:A127923 - cp's OEIS frontend

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

Lekraj Beedassy, Feb 06 2007

nonn

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

Cf. A127923.

Extended by Ray Chandler, Apr 11 2010

cp's OEIS Frontend

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

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