Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations

TL;DR

This paper explores the relationship between multiples of triangular numbers and solutions to Pell equations, providing algebraic expressions and conditions that generate infinite solutions, revealing unexpected links between Pell solutions and constants in the problem.

Contribution

It introduces a novel approach using Pell equations to find and characterize multiples of triangular numbers, including explicit algebraic formulas for fundamental solutions.

Findings

Infinite solutions for multiples of triangular numbers are linked to Pell equation solutions.

Conditions on Pell equation solutions determine which solutions correspond to triangular numbers.

Constants in the problem are directly related to fundamental solutions of Pell equations.

Abstract

For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell equations. With some conditions on parities of fundamental solutions of the simple and generalized Pell equations, only odd solutions of the generalized Pell equation are retained to provide many infinitely solutions found on branches corresponding to each of the generalized fundamental solutions. General algebraic expressions of fundamental solutions of the Pell equations are found for some values of the multiplier k in function of the closest natural square. Further, among the expressions of Pell equation solutions, a set of recurrent relations is identical to those found previously without the Pell equation solving method. It is found also that two constants of the problem of multiples of triangular numbers are directly related to the fundamental solutions of the simple Pell equation, which is an unexpected result as it means that simple Pell equation fundamental solutions in all generality are related to constants in recurrent relations of the problem of finding triangular numbers multiple of other triangular numbers.