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<h2>Rational zeros of a function of three variables.</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 180</h3><span style="width:300px;color:#666;">Published on Saturday, 2nd February 2008, 09:00 am; Solved by 827</span></div>
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<p>For any integer <var>n</var>, consider the three functions</p>
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<p style="margin-left:50px;"><var>f</var><sub>1,<var>n</var></sub>(<var>x</var>,<var>y</var>,<var>z</var>) = <var>x</var><sup><var>n</var>+1</sup> + <var>y</var><sup><var>n</var>+1</sup> <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> <var>z</var><sup><var>n</var>+1</sup><br />
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<var>f</var><sub>2,<var>n</var></sub>(<var>x</var>,<var>y</var>,<var>z</var>) = (<var>xy</var> + <var>yz</var> + <var>zx</var>)*(<var>x</var><sup><var>n</var>-1</sup> + <var>y</var><sup><var>n</var>-1</sup> <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> <var>z</var><sup><var>n</var>-1</sup>)<br />
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<var>f</var><sub>3,<var>n</var></sub>(<var>x</var>,<var>y</var>,<var>z</var>) = <var>xyz</var>*(<var>x</var><sup><var>n</var>-2</sup> + <var>y</var><sup><var>n</var>-2</sup> <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> <var>z</var><sup><var>n</var>-2</sup>)</p>
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<p>and their combination</p>
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<p style="margin-left:50px;"><var>f</var><sub><var>n</var></sub>(<var>x</var>,<var>y</var>,<var>z</var>) = <var>f</var><sub>1,<var>n</var></sub>(<var>x</var>,<var>y</var>,<var>z</var>) + <var>f</var><sub>2,<var>n</var></sub>(<var>x</var>,<var>y</var>,<var>z</var>) <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> <var>f</var><sub>3,<var>n</var></sub>(<var>x</var>,<var>y</var>,<var>z</var>)</p>
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<p>We call (<var>x</var>,<var>y</var>,<var>z</var>) a golden triple of order <var>k</var> if <var>x</var>, <var>y</var>, and <var>z</var> are all rational numbers of the form <var>a</var> / <var>b</var> with<br />
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0 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>a</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>b</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>k</var> and there is (at least) one integer <var>n</var>, so that <var>f</var><sub><var>n</var></sub>(<var>x</var>,<var>y</var>,<var>z</var>) = 0.</p>
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<p>Let <var>s</var>(<var>x</var>,<var>y</var>,<var>z</var>) = <var>x</var> + <var>y</var> + <var>z</var>.<br />
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Let <var>t</var> = <var>u</var> / <var>v</var> be the sum of all distinct <var>s</var>(<var>x</var>,<var>y</var>,<var>z</var>) for all golden triples (<var>x</var>,<var>y</var>,<var>z</var>) of order 35.<br /> All the <var>s</var>(<var>x</var>,<var>y</var>,<var>z</var>) and <var>t</var> must be in reduced form.</p>
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<p>Find <var>u</var> + <var>v</var>.</p>
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