Partitions into Triples with Equal Products and Families of Elliptic Curves

Let Sₗ(M,N) denote a set of triples of positive integers whose parts have the same sum M and the same product N. For each 2≤≤ 4 we establish a connection between the set Sₗ(M,N) and a family of elliptic curves. When =2 and $3$, we use certain known parametrised sets Sₗ(M,N) and respectively find families of elliptic curves of generic rank~≥ 5 and ≥ 6, while for =4 we first obtain a parametrised set Sₗ(M,N), then construct a family of elliptic curves of generic rank ≥ 8. Finally, we perform a computer search within these families to find specific curves with rank~≥ 11. The highest rank examples we found were two curves of rank~$14$.