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

Abstract

A number $N$ is a triangular number if it can be written as $N = t(t + 1)/2$ for some nonnegative integer number $t$. A triangular number $N$ is called square if it is a perfect square, that is, $N = d^2$ for some integer number $d$. Square triangular numbers were characterized by Euler in 1778 and are in one-to-one correspondence with the so-called near-isosceles Pythagorean triples $(k,k+1,l)$, where $k^2 + (k+1)^2 = l^2$. A quadratic number is the product $\Pi = \Pi(k,j) = k(k+1)(k+j)(k+j+1)$ for some nonnegative integer numbers $k$ and $j$. By definition, it is the product of two triangular numbers and 4. Quadratic number $\Pi$ and the corresponding pair $(k,j)$ are called square if $\Pi$ is a perfect square. Clearly, $(k,j)$ is square if both triangular numbers $k(k+1)/2$ and $(k+j)(k+j+1)/2$ are perfect squares. Yet, there exist infinitely many other square quadratic numbers. We construct polynomials $j_i(k)$ of degree $i$ with positive integer coefficients satisfying equations: $k + j_{2 \ell}(k) + 1 = k [a_\ell k^\ell + \dots + a_1 k + a_0]^2 +1 = (k+1) [b_\ell k^\ell + \dots + b_1 k + b_0]^2$ and \newline $k + j_{2\ell+1}(k) + 1 = k(k+1) [a_\ell k^\ell + \dots + a_1 k + a_0]^2 + 1 = [b_{\ell+1} k^{\ell+1}+b_\ell k^\ell + \dots + b_1 k + b_0]^2$ for some positive integer $\ell$ and some coefficients $a_i, b_j$, $i=0, \ldots, \ell, j=0, \ldots, \ell+1$. All the obtained pairs $(k, j_i(k))$ are square. We conjecture that the products of square triangular numbers and pairs $(k, j_i(k))$ cover all quadratic squares. Additionally, we identify pairs of intervals of successive integers with equal sums of squares.