Integer Solutions, Rational solutions of the equations x⁴+y⁴+z⁴ -2(x²)(y²)-2(y²)(z²)-2(z²)(x²)=n and x²+y⁴+z⁴-2x(y²)-2x(z²)-2(y²)(z²)=n; And Crux Mathematicorum Contest problem CC24

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The subject matter of this work are the two equations: x⁴+y⁴+z⁴-2(x²)(y²)-2(y²)(z²)-2(z²)(x²)= n (1) And x²+y⁴+z⁴-2x(y²)-2x(z²)-2(y⁴)(z⁴)= 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=p², 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).