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%I #12 Jun 11 2015 05:36:20
%S 1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,
%T 0,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0,0,1,0,1,
%U 0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0
%N Characteristic triangle for primitive Pythagorean triples.
%C For primitive Pythagorean triples (x,y,z) see the Niven et al. reference, Theorem 5.5, p. 232, and the Hardy-Wright reference, Theorem 225, p.190.
%C T(n, m) = 1 if and only if there exists a primitive Pythagorean triple (x, y, z) with even y, namely x = n^2 - m^2, y = 2*n*m and z = n^2 + m^2.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
%D Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.
%F T(n, m) = 1 if n > m >= 1, (-1)^(n+m) = -1 and gcd(n, m) = 1, else 0.
%e The triangle T(n, m) begins:
%e n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
%e --------------------------------------------------
%e 2: 1
%e 3: 0 1
%e 4: 1 0 1
%e 5: 0 1 0 1
%e 6: 1 0 0 0 1
%e 7: 0 1 0 1 0 1
%e 8: 1 0 1 0 1 0 1
%e 9: 0 1 0 1 0 0 0 1
%e 10: 1 0 1 0 0 0 1 0 1
%e 11: 0 1 0 1 0 1 0 1 0 1
%e 12: 1 0 0 0 1 0 1 0 0 0 1
%e 13: 0 1 0 1 0 1 0 1 0 1 0 1
%e 14: 1 0 1 0 1 0 0 0 1 0 1 0 1
%e 15: 0 1 0 1 0 0 0 1 0 0 0 0 0 1
%e ...
%e T(5, 2) = 1 because the Pythagorean triple (21, 20, 29) is primitive (pairwise coprime).
%e T(5, 3) = 0 because the Pythagorean triple (16, 30, 34) is not primitive.
%o (PARI) {T(n,m) = n>m && m>0 && (n+m)%2 && gcd(n,m) ==1}; /* _Michael Somos_, Dec 05 2014 */
%Y Cf. A208853, A208854, A208855, A222946, A225949, A225950, A225952.
%K nonn,easy,tabl
%O 2
%A _Wolfdieter Lang_, Dec 03 2014
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