Consecutive numbers in multiplication tables

ABSTRACT A multiplication table of size $x$ is the set of all possible products $a \cdot b$ with $1 \le a, b \le \sqrt{x}$. In this article, we study the number of pairs of consecutive integers in a multiplication table and obtain non-trivial lower and upper bounds of its size. For multiplication rectangle with side lengths $x^{0.088}$ and $x^{0.912}$, we obtain an almost optimal asymptotic, showing evidence of independence of integers lying in multiplication tables.