On the Diophantine Equation x6 + ky3 = z6 + kw3

Given the positive integers m; n, solving the well known sym- metric Diophantine equation xm+kyn = zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers 1 ≤ k ≤ 500 the Diophantine equation x6 + ky3 = z6 + kw3 has innitely many nontrivial (y 6= w and x 6= z) 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 innitely many) rational numbers k for which this Diophantine equation is satised. Fi- nally, appealing these observations we conjecture that the above result is true for all rational numbers k.