[seqfan] Two elusive triangular numbers

[mathstutoring]

The triangular numbers T(14)=105 and T(18)=171 have the property that their sum, difference and product are all triangular numbers.

Does anyone know of the existence of a pair of triangular numbers T(m) and T(n) such that their sum, difference, product and quotient are also triangular numbers.

Or alternatively know of the existence of a proof that no such m and n exist.

Ant