Integer Solutions, Rational Solutions of the Equations x4+y4+z4-2x2y2-2y2z2-2z2x2=n and x2+y4+z4-2xy2-2xz2-2y2z2=n - And Crux Mathematicorum Contest Corner Problem CC24
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Crux Mathematicorum, No 5 (May 2013), Vol 38, Contest Corner problem CC24, page 174 Given the equation, x4 y 4 z 4 2 y 2 z 2 2z 2 x2 2 x2 y 2 24 (a) Prove that the equation has no integer solutions (b) Does this equation have rational solutions? If yes, give an example. If no, prove it.
W. Sierpinski, Elementary Theory of Numbers, Warsaw 1964, 480pp. ISBN: 0-598-52758-3 For the result that states that is a natural number is the mth power of rational number, and m is natural; see Theorem 7, on page 16 For the Pythagorean equation x2 y 2 z 2 , see pages 3842.
Konstantine Zelator, The Diophantine equation x2 ky2 z 2 , and Integral Triangles with a Cosine Value of Mathematics and Computer Education Vol 30, No 3, pp.191-197, Fall (2006).