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.