Abstract:We prove that
$164\, 634\, 913$
is the smallest positive integer that is a sum of two rational sixth powers, but not a sum of two integer sixth powers. If
$C_{k}$
is the curve
$x^{6} + y^{6} = k$
, we use the existence of morphisms from
$C_{k}$
to elliptic curves, together with the Mordell–Weil sieve, to rule out the existence of rational points on
$C_{k}$
for var… Show more
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