Some Remarks on Triangular Numbers

Chahal, Jasbir S.; D’Souza, Harry

doi:10.4324/9780203747018-9

Cited by 1 publication

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“…As an application of the above discussion of the Pell equation, we recall from [ChaD93] the following generalization of Euler's work regarding which triangular numbers are three times some other triangular numbers.…”

Section: A Triangular Numbersmentioning

confidence: 99%

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When are Multiples of Polygonal Numbers again Polygonal Numbers?

Chahal

Griffin²,

Priddis³

2019

Hardy-Ramanujan Journal

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International audience 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.

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Section: A Triangular Numbersmentioning

confidence: 99%

“…Given an integer m > 1, it is interesting to study which triangular numbers are m times another triangular number. In this paper, we recall the answer to this question (see [ChaD93]) and determine, more generally, which -gonal numbers are m times another -gonal number.…”

Section: Introductionmentioning

confidence: 99%