## Osaka Journal of Mathematics

### Partitions with equal products and elliptic curves

#### Abstract

Let $a$, $b$, $c$ be distinct positive integers. Set $M = a+b+c$ and $N = abc$. We give an explicit description of the Mordell--Weil group of the elliptic curve $E_{(M, N)}\colon y^{2}-Mxy-Ny = x^{3}$ over $\mathbb{Q}$. In particular we determine the torsion subgroup of $E_{(M, N)}(\mathbb{Q})$ and show that its rank is positive. Furthermore there are infinitely many positive integers $M$ that can be written in $n$ different ways, $n\in\{2, 3\}$, as the sum of three distinct positive integers with the same product $N$ and $E_{(M, N)}(\mathbb{Q})$ has rank at least $n$.

#### Article information

Source
Osaka J. Math., Volume 52, Number 2 (2015), 515-527.

Dates
First available in Project Euclid: 24 March 2015

https://projecteuclid.org/euclid.ojm/1427202900

Mathematical Reviews number (MathSciNet)
MR3326624

Zentralblatt MATH identifier
1317.14072

#### Citation

Sadek, Mohammad; El-Sissi, Nermine. Partitions with equal products and elliptic curves. Osaka J. Math. 52 (2015), no. 2, 515--527. https://projecteuclid.org/euclid.ojm/1427202900

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