On the difference between squares and integral $S$-units

TL;DR

This paper addresses lower bounds for the S-part of $|x^2 - q_1^{a_1}\cdots q_t^{a_t}|$ with disjoint prime sets $S$ and $T$, and derives consequences for the greatest prime factor of the difference. It blends Schinzel’s method with sharp linear forms in logarithms, both Archimedean and $p$-adic, along with $S$-unit techniques to obtain two bounds: an ineffective exponent $\frac{1}{2}+\varepsilon$ and an effective exponent $1-\kappa$, where $\kappa = \bigl(c^s (\log\log P) (\prod_{i=1}^s \log p_i)^2 \bigr)^{-1}$ and $P = \max\{p_1,\dots,p_s\}$. The results yield a quantitative lower bound for the greatest prime factor $P[ x^2 - q_1^{a_1}\cdots q_t^{a_t} ]$, and a growth bound $P[ \cdot ] \gg^{\mathrm{eff}}_T \log_* \log X$ with further refinements, including connections to higher powers and related $S$-unit problems. The techniques have potential applications to Diophantine equations of the form $x^2 - q_1^{a_1} \cdots q_t^{a_t} = y^n$ and to broader $S$-unit extremal questions.

Abstract

Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers and $x$ a positive integer. We establish an effective lower bound for the greatest prime divisor of $|x^2 - q_1^{a_1} \ldots q_t^{a_t}|$, which tends to infinity with the maximum of $x$, $a_1, \ldots , a_t$.

Theorems & Definitions (12)

• Theorem 1.1: Schinzel, 1967
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• proof