On Polygonal Square Triangular Numbers

A pentagonal square triangular number is a number which is a pentagonal number P5(ℓ), a square y2 and a triangular number P3(m) at the same time. It would be well known for the specialists that there exists no pentagonal square triangular number except for P3(1) = 12 = P5(1) = 1. But we don’t know any simple reference of the proof of this fact in print. The object of this note is to provide a such reference. Here we shall present three independent proofs of this fact one of which was already referred in the net article [24].