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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.15365 |
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Table of Contents:
- In this paper we study the problem of long gaps between values of binary quadratic forms. Let $D_{1}$, $D_{2},\ldots ,D_{r}$ be negative integers and $(s_{n})_{n=1}^{\infty}$ be the sequence of all the numbers representable by any binary quadratic form of discriminant $D_{1}$, $D_{2}$, $\ldots$ or $D_{r}$, and let $d :={\rm lcm}\{D_{1},\ldots ,D_{r}\}$. We show that then \begin{align*} \limsup_{n\to\infty}\frac{s_{n+1}-s_{n}}{\log s_{n}}\geq \frac{1}{\log d + \log\log d + \log\log\log d + 4}. \end{align*} This improves and generalises a result by Dietmann, Elsholtz, Kalmynin, Konyagin, and Maynard. As a by-product of our preliminary results, we show an improvement to the Pólya-Vinogradov inequality.