HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation
Journal articles

When are Multiples of Polygonal Numbers again Polygonal Numbers?

Abstract : 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.
Document type :
Journal articles
Complete list of metadata

Cited literature [18 references]  Display  Hide  Download

Contributor : Srinivas Kotyada Connect in order to contact the contributor
Submitted on : Friday, January 18, 2019 - 8:06:31 PM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM


Files produced by the author(s)




Jasbir Chahal, Michael Griffin, Nathan Priddis. When are Multiples of Polygonal Numbers again Polygonal Numbers?. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2019, Atelier Digit_Hum, pp.58 - 67. ⟨10.46298/hrj.2019.5107⟩. ⟨hal-01986591⟩



Record views


Files downloads