## 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

#### Abstract

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

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

Dates
First available in Project Euclid: 28 March 2018

https://projecteuclid.org/euclid.facm/1522202458

Digital Object Identifier
doi:10.7169/facm/1700

Mathematical Reviews number (MathSciNet)
MR3932606

Zentralblatt MATH identifier
07055566

#### Citation

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. https://projecteuclid.org/euclid.facm/1522202458

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