A127924 - OEIS

%I #5 Mar 31 2012 10:26:06

%S 1,17,23,103,137,199,217,497,601,697,799,1343,1457,1679,1799,2737,

%T 2839,2921,3199,3337,3503,3503,3937,3961,3977,4183,4577,4657,5543,

%U 6103,6463,7399,7663,8143,8977,9143,9881,10097,10577,10897,10943,11543,13703,13817

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

%C 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.

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

%Y Cf. A127923.

%K nonn

%O 1,2

%A _Lekraj Beedassy_, Feb 06 2007

%E Extended by _Ray Chandler_, Apr 11 2010