On the Diophantine Inequality $|x^2-2^a\cdot 3^b|<3\max \{a,b\}$

Abstract

In this paper, we show that there are $57$ nonnegative integer solutions $(a,b,x)$ to the inequality $1\le |x^2-2^a\cdot 3^b|<3\max \{a,b\}$ and we list them explicitly. The inequality is converted into a statement about how closely $x/q$ approximates irrational number $\sqrt{d}$ for $d\in \{2,3,6\}$, where $q$ is an integer which is $3$-smooth, after which Worley's theorem on rational approximations via continued fractions is applied to parametrise the solutions and a $p$-adic lower bound for a linear form in logarithms due to Bugeaud and Laurent is applied to find a rather large bound on $\max \{a,b\}$. We finish with an application of the LLL algorithm to reduce this bound.

17 pages