Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers

TL;DR

This paper investigates the congruence properties of indices of triangular numbers that are multiples of other triangular numbers, revealing patterns and exceptions that optimize numerical searches for solutions.

Contribution

It provides a detailed analysis of the congruence relations of indices, including new algebraic expressions and rules for various cases, improving search efficiency.

Findings

Remainders in congruence relations always come in pairs summing to (k-1).

Special sets of remainders include 0, (k-1), n, and (n^2 - 1) under certain conditions.

The approach reduces the search space for solutions by eliminating impossible index values.

Abstract

It is known that, for any positive non-square integer multiplier $k$, there is an infinity of multiples of triangular numbers which are triangular numbers. We analyze the congruence properties of the indices $\xi$ of triangular numbers that are multiples of other triangular numbers. We show that the remainders in the congruence relations of $\xi$ modulo k come always in pairs whose sum always equal $\left(k-1\right)$, always include 0 and $\left(k-1\right)$, and only 0 and $\left(k-1\right)$ 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 $\left(n^{2}-1\right)$ and if $k$ has integer factors, the set of remainders include multiples of a factor following certain rules. Finally, algebraic expressions are found for remainders in function of $k$ and its factors. Several exceptions are noticed and superseding rules exist between various rules and expressions of remainders. This approach allows to eliminate in numerical searches those $\left(k-\upsilon\right)$ values of $\xi_{i}$ that are known not to provide solutions, where $\upsilon$ is the even number of remainders. The gain is typically in the order of $k/\upsilon$, with $\upsilon\ll k$ for large values of $k$.