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
TL;DR
The paper investigates when the quadratic number $\Pi(k,j)=k(k+1)(k+j)(k+j+1)$ is a perfect square and constructs, for each $i$, a polynomial $j_i(k)$ with positive integer coefficients such that $\Pi(k,j_i(k))$ is a square for all $k$. It links square triangular numbers to near-isosceles Pythagorean triples and Fermat–Pell equations, developing both recursive and explicit parametrizations. The authors prove the existence of a unique triplet of polynomials $A_i,B_i,C_i$ yielding identities that force square quadratic numbers, and conjecture that these families exhaust all such squares, while also connecting the problem to pairs of intervals with equal sums of squares. They extend the study to generalizations with $s$-vectors and provide extensive computational data for small parameters, highlighting rich diophantine structure and open questions. The work contributes new structured parametrizations and conjectures tying classical number-theoretic objects to modern polynomial constructions and interval-sum identities with potential SEO and downstream mathematical applications.
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 $Π= Π(k,j) = k(k+1)(k+j)(k+j+1)$ for some nonnegative integer numbers $k$ and $j$. By definition, it is the product of two triangular numbers and 4. Quadratic number $Π$ and the corresponding pair $(k,j)$ are called square if $Π$ 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.