Question

Express 39312 as a sum of two cubes in two different ways

Answer 1

Answer:

It seems known that there are infinitely many numbers that can be expressed as a sum of two positive cubes in at least two different ways (per the answer to this post: Number Theory Taxicab Number).

We know that

1729=103+93=123+13,

1729=103+93=123+13,

and I am wondering if there are infinitely many numbers like this that can be expressed as the sum of two positive cubes in exactly two ways?

In fact, are there even any other such numbers?

EDIT: As provided by MJD in the comments section, here are other examples:

4104=23+163=93+153,

4104=23+163=93+153,

13832=203+183=243+23,

13832=203+183=243+23,

20683=103+273=193+243.