## Rational zeros of a function of three variables

### Problem 180

Published on Saturday, 2nd February 2008, 09:00 am; Solved by 898For any integer `n`, consider the three functions

`f`_{1,n}(`x`,`y`,`z`) = `x`^{n+1} + `y`^{n+1} − `z`^{n+1}

`f`_{2,n}(`x`,`y`,`z`) = (`xy` + `yz` + `zx`)*(`x`^{n-1} + `y`^{n-1} − `z`^{n-1})

`f`_{3,n}(`x`,`y`,`z`) = `xyz`*(`x`^{n-2} + `y`^{n-2} − `z`^{n-2})

and their combination

`f`_{n}(`x`,`y`,`z`) = `f`_{1,n}(`x`,`y`,`z`) + `f`_{2,n}(`x`,`y`,`z`) − `f`_{3,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 `f`_{n}(`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`.