Partitions with equal products. II

Abstract:

The following theorem is proved: Let $k \geq 3$ and $r$ be positive integers. There exist infinitely many integers having $r$ partitions into $k$ parts such that the products of the integers in each partition are equal. Moreover, these partitions are mutually disjoint, i.e., no integer occurs in more than one of them. Of some additional interest is a lemma stating that a certain class of elliptic curves has positive rank over ${\mathbf {Q}}$.