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

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).