arXiv subjects
Publications and source records attributed to Sanda Bujačić Babić.
We prove that every Diophantine quadruple $\{a,b,c,d\}$ over $\mathbb{Q}(i)[X]$ that contains at least one non-constant polynomial is regular; that is, $$(a+b-c-d)^2=4(ab+1)(cd+1).$$ This result is consistent with the corresponding results over $\mathbb{Z}[i][X]$ and $\mathbb{R}[X]$, but contrasts with the situation over $\mathbb{C}[X]$.
A set $\{a, b, c, d\}$ of four non-zero distinct polynomials in $\mathbb{Z}[i][X]$ is said to be a Diophantine $D(4)$-quadruple if the product of any two of its distinct elements increased by 4 is a square of some polynomial in $\mathbb{Z}[i][X]$. In this paper we prove that every $D(4)$-quadruple in $\mathbb{Z}[i][X]$ is regular, or equivalently that the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds for every $D(4)$-quadruple in $\mathbb{Z}[i][X]$.
We prove there exist infinitely many odd integers $n$ for which there exists a pair of positive divisors $d_1, d_2>1$ of $(n^2+1)/2$ such that $$d_1+d_2=δn+(δ+2).$$ We prove the similar result for $\varepsilon=δ-2$ and $δ\equiv4, 6\pmod{8}$ using different approaches and methods.