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

A number $N$ is a triangular number if it can be written as $N=t(t+1)\mathrm{/}2$ 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 $\mathrm{\Pi }=\mathrm{\Pi }(k,j)=k(k+1)(k+j)(k+j+1)$ for some nonnegative integer numbers $k$ and $j$. By definition, it is the product of two triangular numbers and 4. Quadratic number $\mathrm{\Pi }$ and the corresponding pair $(k,j)$ are called square if $\mathrm{\Pi }$ is a perfect square. Clearly, $(k,j)$ is square if both triangular numbers $k(k+1)\mathrm{/}2$ and $(k+j)(k+j+1)\mathrm{/}2$ 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: $k+j_{2\mathrm{\ell }}(k)+1=k[a_{\mathrm{\ell }}{k}^{\mathrm{\ell }}+\cdots +a_{1}k+a_0{]}^2+1=(k+1)[b_{\mathrm{\ell }}{k}^{\mathrm{\ell }}+\cdots +b_{1}k+b_0{]}^2$ and \newline $k+j_{2\mathrm{\ell }+1}(k)+1=k(k+1)[a_{\mathrm{\ell }}{k}^{\mathrm{\ell }}+\cdots +a_{1}k+a_0{]}^2+1=[b_{\mathrm{\ell }+1}{k}^{\mathrm{\ell }+1}+b_{\mathrm{\ell }}{k}^{\mathrm{\ell }}+\cdots +b_{1}k+b_0{]}^2$ for some positive integer $\mathrm{\ell }$ and some coefficients $a_i,b_j$, $i=0,\dots ,\mathrm{\ell },j=0,\dots ,\mathrm{\ell }+1$. 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.