Hardy Ramanujan Journal |

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation ∆ = m∆' is satisfied by infinitely many pairs of triangular numbers ∆, ∆'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit of Q(√ m), finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP' , P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P , P' , P'' of polygonal numbers, and give examples of such solutions.

Source : oai:HAL:hal-01986591v1

Published on: January 23, 2019

Submitted on: January 23, 2019

Keywords: polygonal numbers,Pell equation,triangular numbers,elliptic curves,Diophantine equations 2010 Mathematics Subject Classification primary:11D45,secondary:11G05,
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MATH
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Mathematics [math],
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MATH.MATH-NT
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Mathematics [math]/Number Theory [math.NT]

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