Functiones et Approximatio Commentarii Mathematici

A note on the Diophantine equation $2^{n-1}(2^{n}-1)=x^3+y^3+z^3$

Maciej Ulas

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Motivated by a recent result of Farhi we show that for each $n\equiv \pm 1\pmod{6}$ the title Diophantine equation has at least two solutions in integers. As a consequence, we get that each (even) perfect number is a sum of three cubes of integers. Moreover, we present some computational results concerning the considered equation and state some questions and conjectures.

Article information

Funct. Approx. Comment. Math., Volume 60, Number 1 (2019), 87-96.

First available in Project Euclid: 28 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas 11B13: Additive bases, including sumsets [See also 05B10]

perfect numbers sums of three cubes


Ulas, Maciej. A note on the Diophantine equation $2^{n-1}(2^{n}-1)=x^3+y^3+z^3$. Funct. Approx. Comment. Math. 60 (2019), no. 1, 87--96. doi:10.7169/facm/1700.

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