Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers
Abstract
It is known that, for any positive nonsquare 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 $\left(k1\right)$, always include 0 and $\left(k1\right)$, and only 0 and $\left(k1\right)$ 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 $k$.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2103.03019
 Bibcode:
 2021arXiv210303019P
 Keywords:

 Mathematics  General Mathematics;
 11A25 (Primary);
 11D09 (Secondary)
 EPrint:
 9 pages. arXiv admin note: text overlap with arXiv:2102.13494