Integer Solutions, Rational solutions of the equations x^4+y^4+z^4 -2(x^2)(y^2)-2(y^2)(z^2)-2(z^2)(x^2)=n and x^2+y^4+z^4-2x(y^2)-2x(z^2)-2(y^2)(z^2)=n; And Crux Mathematicorum Contest problem CC24

TL;DR

This paper investigates integer and rational solutions of two complex polynomial equations related to a contest problem, providing new theorems that classify solutions based on the parameter n and generalize previous results.

Contribution

The paper offers new theorems that determine the existence and classification of solutions for the equations, extending prior work and solving open questions from the contest problem.

Findings

Equation (1) has no solutions when n=8N with N odd.

Equation (2) has no integer solutions when n is prime, 4, or product of two distinct primes.

All solutions are classified for specific forms of n, including prime squares and certain congruences.

Abstract

The subject matter of this work are the two equations: x^4+y^4+z^4-2(x^2)(y^2)-2(y^2)(z^2)-2(z^2)(x^2)= n (1) And x^2+y^4+z^4-2x(y^2)-2x(z^2)-2(y^4)(z^4)= n (2) where n is a natural number. Contest Corner problem CC24, published in the May2012 issue of the journal Crux Mathematicorum(see reference[1]); provided the motivation behind this work. In Th.1, we show that eq.(1) if n=8N, N odd; then eq.(1) has no integer solutions; which generalizes problem CC24(the case n=24). We use Th.2, to find some rational solutions of eq.(1); which answers the second question in CC24. In Th.4, we show that if n= p, 4, or pq; where p and q are distinct primes. Then eq.(1)has no integer solutions. In Th.6, we determine all the integer solutions to (1), when n=p^2, p an odd prime. Theorems 7 through13, deal with equation (2). In Th.11, we determine all the integer solutions of eq.(2). Th.12 states that (2) has no integer solutions in n is congruent to 2 or 3 modulo4. Finally, in Th.13 we determine all the rational solutions of eq.(2).