Large Gaps between Sums of Two Squareful Numbers

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-07-05 DOI:10.1134/s000143462403026x

A. B. Kalmynin, S. V. Konyagin

引用次数: 0

Abstract

Let $M(x)$ be the length of the largest subinterval of $[1,x]$ which does not contain any sums of two squareful numbers. We prove a lower bound

$$M(x)\gg \frac{\ln x}{(\ln\ln x)^2}$$

for all $x\geq 3$. The proof relies on properties of random subsets of the prime numbers.

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两个平方数之和之间的巨大差距

摘要让$M(x)$是$[1,x]$的最大子区间的长度，它不包含任何两个平方数的和。我们为所有的 $x\geq 3$ 证明了一个下界 $$M(x)\gg \frac{ln x}{(\ln\ln x)^2}$$ 。证明依赖于素数随机子集的性质。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.