Given the positive integers m,n, solving the well known symmetric Diophantine equation x^{m}+ky^{n}=z^{m}+kw^{n}, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x^{6}+ky^{3}=z^{6}+kw^{3} has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is not greater than 500. We exhibit a collection of (probably infinitely many) rational numbers k for which this Diophantine equation is satisfied. Finally, appealing these observations, we conjecture that the above result is true for all rational numbers k.

Type of Study: Research paper |
Subject:
Special

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