freeCodeCamp/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-180-rational-zeros-of-a-function-of-three-variables.md

id, title, challengeType, forumTopicId, dashedName

id	title	challengeType	forumTopicId	dashedName
5900f4201000cf542c50ff33	Problem 180: Rational zeros of a function of three variables	5	301816	problem-180-rational-zeros-of-a-function-of-three-variables

--description--

For any integer n, consider the three functions

$$\begin{align} & f_{1,n}(x,y,z) = x^{n + 1} + y^{n + 1} − z^{n + 1}\\ & f_{2,n}(x,y,z) = (xy + yz + zx) \times (x^{n - 1} + y^{n - 1} − z^{n - 1})\\ & f_{3,n}(x,y,z) = xyz \times (x^{n - 2} + y^{n - 2} − z^{n - 2}) \end{align}$$

and their combination

$$\begin{align} & f_n(x,y,z) = f_{1,n}(x,y,z) + f_{2,n}(x,y,z) − f_{3,n}(x,y,z) \end{align}$$

We call (x,y,z) a golden triple of order k if x, y, and z are all rational numbers of the form \frac{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 = \frac{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.

--hints--

rationalZeros() should return 285196020571078980.

assert.strictEqual(rationalZeros(), 285196020571078980);

--seed--

--seed-contents--

function rationalZeros() {

  return true;
}

rationalZeros();

--solutions--

// solution required