Partitions with equal products and elliptic curves

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 $\displaystyle E_{(M,N)}:y^2-Mxy-Ny=x^3$ over $\Q$. In particular we determine the torsion subgroup of $E_{(M,N)}(\Q)$ 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 $E_{(M,N)}(\Q)$ has rank at least $n$.

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