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Rational zeros of a function of three variables

Problem 180

Published on 02 February 2008 at 09:00 am [Server Time]

For any integer n, consider the three functions

f1,n(x,y,z) = xn+1 + yn+1 − zn+1
f2,n(x,y,z) = (xy + yz + zx)*(xn-1 + yn-1 − zn-1)
f3,n(x,y,z) = xyz*(xn-2 + yn-2 − zn-2)

and their combination

fn(x,y,z) = f1,n(x,y,z) + f2,n(x,y,z) − f3,n(x,y,z)

We call (x,y,z) a golden triple of order k if x, y, and z are all rational numbers of the form a / b with
0 < a < b ≤ k and there is (at least) one integer n, so that fn(x,y,z) = 0.

Let s(x,y,z) = x + y + z.
Let t = u / v be the sum of all distinct s(x,y,z) for all golden triples (x,y,z) of order 35.
All the s(x,y,z) and t must be in reduced form.

Find u + v.


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