Representing Even Perfect and Near-Perfect Numbers as Sums of Cubes
Abstract
Motivated by recent results of Farhi and Ulas we show that for each $n\equiv \pm 1 \pmod 6$, the Diophantine equation $2^{n-1}(2^n-1)=x^3+y^3+z^3$ has at least three solutions. We also prove results about the representation of even near-perfect numbers with two distinct prime factors as sums of integral cubes.
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