

A127924


Oneseventh of the difference of squares of legs of primitive Pythagorean triangles, neither of which is a multiple of 7.


0



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
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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


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KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



