Congruence properties of indices of triangular numbers multiple of other triangular numbers

Vladimir Pletser

doi:10.30538/oms2021.0162

Abstract

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple of other triangular numbers. We analyze the congruence properties of indices \(\xi\) of triangular numbers multiple of triangular numbers. Remainders in congruence relations \(\xi\) modulo \(k\) come always in pairs whose sum always equal \((k-1)\), always include 0 and \((k-1)\), and only 0 and \((k-1)\) if \(k\) is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier \(k\) is twice the triangular number of \(n\), the set of remainders includes also \(n\) and \((n^{2}-1)\) and if \(k\) has integer factors, the set of remainders include multiples of a factor following certain rules. Algebraic expressions are found for remainders in function of \(k\) and its factors, with several exceptions. This approach eliminates those \(\xi\) values not providing solutions.

Highlights

Triangular numbers Tt =t(t+1) 2 are one of the figurate numbers enjoying many properties; see, e.g., [1,2]for relations and formulas
In [9], we showed that the rank r is the index of tr and ξr solutions of (1) such that κ = tr + tr−1 = ξr − ξr−1 − 1, (2)
For indices ξ of triangular numbers multiples of other triangular numbers, the remainders in the congruence relations of ξ modulo k always come in pairs whose sum always equal (k − 1), always include 0 and (k − 1), and only 0 and (k − 1) if k is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two

Summary

Introduction

Pletser [9] showed that, for non-square integer values of k, there are infinitely many solutions that can be represented by recurrent relations of the four variables t, ξ, Tt and Tξ, involving a rank r and parameters κ and γ, which are respectively the sum and the product of the (r − 1)th and the rth values of t. We present a method based on the congruent properties of ξ (mod k), searching for expressions of the remainders in function of k or its factors. This approach accelerates the numerical search of the values of tn and ξn that solve (1), as it eliminates values of ξ that are known not to provide solutions to (1). The gain is typically in the order of k/υ where υ is the number of remainders, which is usually such that υ k

Rank and Recurrent Equations

Congruence of ξ modulo k

Conclusions