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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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

Permanent link to this document
https://projecteuclid.org/euclid.facm/1522202458

Digital Object Identifier
doi:10.7169/facm/1700

Mathematical Reviews number (MathSciNet)
MR3932606

Zentralblatt MATH identifier
07055566

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

Keywords
perfect numbers sums of three cubes

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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References

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