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 The smallest sum (Posted on 2012-06-02)
Find three distinct integers, X, Y and Z, such that X + Y, X + Z, Y + Z, X - Y, X - Z, and Y - Z are all squares of integers.
Apparently, there are many solutions.

Find the set [X, Y, Z] with the smallest X + Y + Z.

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 Some thoughts (spoiler) Comment 8 of 8 |
With the restriction that Z = -Y, the necessary squares are reduced to three in number; namely X + Y, X – Y and 2Y.

If we write         X + Y = a2,        X – Y = b2          2Y = c2

then                  X = (a2 + b2)/2,              Y = (a2 – b2)/2

and we require that a2 = b2 + c2. We can use standard formulae to generate all coprime Pythagorean triples (a, b, c) using positive integers p, q, k, with p and q coprime and of opposite parity, and with p > q.

Case 1:
a = k(p2 + q2),   b = k(p2 – q2),    c = 2kpq

giving:  X = k2(p4 +q4)  and  Y = 2k2p2q2

For small values of p, q, k, these generate [X,Y] pairs like:
[17,8], [68,32], [97,72], [153,72], [257,32], [272,128], [337,288]...

Case 2:             a = k(p2 + q2),   b = 2kpq,          c = k(p2 – q2)

giving:  X = k2(p4 + q4 + 6p2q2)/2   and  Y = k2(p2 – q2)2/2

Since p and q have opposite parity, in case 2, X and Y will be integers
iff k is even. So we could put k = 2K and for K = 1, 2, .. use the
modified formula:

X = 2K2(p4 + q4 + 6p2q2)   and  Y = 2K2(p2 – q2)2

For small values of p, q, k, these generates [X,Y] pairs like:
[82,18], [328,72], [626,50], [706,450], [738,162], [1312,288]...

If k = 1, only coprime triples (a, b, c) will be generated by the coprime pairs
(p, q). So the parameter k is needed to produce all multiples of these triples.

 Posted by Harry on 2012-06-09 18:43:21

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