Speaker
Description
Let $\mathcal R$ be a commutative ring with unity and $n\in \mathcal R$, $n\not=0$. A $D(n)$-quadruple in $\mathcal R$ is a set of four elements in $\mathcal{R}\backslash\{0\}$ with the property that the product of any two of its distinct elements increased by $n$ is a square in $\mathcal{R}$. It is interesting that in some rings $D(n)$-quadruples are related to the representations of $n$ by the binary quadratic form
$x^2 - y^2$. Moreover, there are many examples of rings of integers of number fields in which a $D(n)$-quadruple exists if and only if $n$ can be written as a difference of two squares in $\mathcal R$. Here we investigate the connection between “D(n)-quadruples and
differences of squares” in the ring of polynomials with integer coefficients, $\mathbb Z[X]$ and show that there is no polynomial $D(n)$-quadruple in $\mathbb Z[X]$ for certain $n\in\mathbb Z[X]$ that are not representable as a difference of squares of two polynomials in
$\mathbb Z[X]$.
