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arXiv · 2508.06598

On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares

Vladimir Gurvich·Mariya Naumova
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Abstract

In 1778 Leonhard Euler characterized triangular numbers that are perfect squares. Obviously, the product of any two such numbers is a perfect square too. Yet, there are many other solutions, that is, pairs $(k,k')$ such that $k(k+1)k'(k'+1)$ is a perfect square. We give explicit formulas characterizing all these square triangular pairs by means of some integer positive polynomials, which is the primary novelty of our work. This result allows us to find all pairs of intervals of successive integers with equal sums of squares in case when the lengths of two intervals in a pair differ by 1. It is known that there is a one-to-one correspondence between the square triangular numbers and nearly isosceles Pythagorean triples: $n^2 + (n+1)^2 = N^2$. Both are generated by the same Fermat-Pell recursion. This is a special case of our result, when the lengths of the two intervals are 2 and 1.

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Vladimir Gurvich, Mariya Naumova. 2026-09-15. On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares. https://arxiv.org/abs/2508.06598

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