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On polygonal square triangular numbers. (English) Zbl 1482.11039

Summary: A pentagonal square triangular number is a number which is a pentagonal number \(P_5(\ell)\), a square \(y^2\) and a triangular number \(P_3(m)\) at the same time. It would be well known for the specialists that there exists no pentagonal square triangular number except for \(P_3(1)=1^2=P_5(1)=1\). 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 [Wolfram Mathworld, Pentagonal Triangular Number, https://mathworld.wolfram.com/PentagonalTriangularNumber.html].

MSC:

11D09 Quadratic and bilinear Diophantine equations
11G05 Elliptic curves over global fields
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