On a conjecture of Erdős

Chen, Yong-Gao; Ding, Yuchen

doi:10.5802/crmath.345

Comptes Rendus. Mathématique

2022

DOI: 10.5802/crmath.345

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On a conjecture of Erdős

Yong-Gao Chen¹,

Yuchen Ding²

Abstract: In this note, we confirm an old conjecture of Erdős.

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Cited by 2 publications

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Quantitative results of the Romanov type representation functions

Chen

Ding

2023

The Quarterly Journal of Mathematics

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For α > 0, let$$\mathscr{A}=\{a_1 \lt a_2 \lt a_3\lt\cdots\}$$and$$\mathscr{L}=\{\ell_1, \ell_2, \ell_3,\cdots\} \quad \text{(not\ necessarily\ different)}$$be two sequences of positive integers with $\mathscr{A}(m)\gt(\log m)^\alpha $ for infinitely many positive integers m and $\ell_m\lt0.9\log\log m$ for sufficiently large integers m. Suppose further that $(\ell_i,a_i)=1$ for all i. For any n, let $f_{\mathscr{A},\mathscr{L}}(n)$ be the number of the available representations listed below$$\ell_in=p+a_i \quad \left(1\le i\le \mathscr{A}(n)\right),$$where p is a prime number. It is proved that$$\limsup_{n\to \infty } \frac{f_{\mathscr{A},\mathscr{L}}(n)}{\log\log n}\gt0,$$which covers an old result of Erdős in 1950 by taking $a_i=2^i$ and $\ell_i=1$. One key ingredient in the argument is a technical lemma established here, which illustrates how to pick out the admissible parts of an arbitrarily given set of distinct linear functions. The proof then reduces to the verifications of a hypothesis involving well-distributed sets introduced by Maynard, which of course would be the other key ingredient in the argument.

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Quantitative results of the Romanov type representation functions

Chen

Ding

2023

The Quarterly Journal of Mathematics

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On the sum of a Fibonacci number and a prime

Wang

Chen

2022

Int. J. Number Theory

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On a conjecture of Erdős

Abstract: In this note, we confirm an old conjecture of Erdős.

Cited by 2 publications

References 8 publications

Quantitative results of the Romanov type representation functions

Quantitative results of the Romanov type representation functions

On the sum of a Fibonacci number and a prime

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