2013/08/16 by Konstantine Zelator, Zelator, Konstantine
Mathematics · #Advanced Differential Equations and Dynamical Systems #math.GM #msc:A005
paper · pdf · doi:10.48550/arxiv.1308.4040
23 pages, no figures
arxiv created 2013/08/16 · arxiv updated 2013/08/20
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).