# On certain ratios regarding integer numbers which are both triangulars and squares

## TL;DR

This paper studies the ratios between successive numbers that are both triangular and perfect squares, revealing new insights about the limits of their ratios and differences, and proposing related conjectures.

## Contribution

It introduces a novel investigation into the limit of the ratio of differences of successive triangular-square numbers, showing it matches the known ratio limit.

## Key findings

- The limit of the ratio of successive numbers is known.
- The paper proves the limit of the ratio of differences coincides with the known ratio.
- A conjecture about related limits is formulated.

## Abstract

We investigate integer numbers which possess at the same time the properties to be triangulars and squares, that are, numbers $a$ for which do exist integers $m$ and $n$ such that $ a = n^2 = \frac{m \cdot (m+1)}{2} $. In particular, we are interested about ratios between successive numbers of that kind. While the limit of the ratio for increasing $a$ is already known in literature, to the best of our knowledge the limit of the ratio of differences of successive ratios, again for increasing $a$, is a new investigation. We give a result for the latter limit, showing that it coincides with the former one, and we formulate a conjecture about related limits.