On a conjecture of Erd\H{o}s

TL;DR

This paper confirms Erdős's longstanding conjecture that for large enough x and a sufficiently large set of integers, there exists an integer n with many representations as n = p + a_i, where p is prime.

Contribution

The paper proves Erdős's conjecture, establishing the existence of an integer with numerous prime sum representations for large sets of integers.

Findings

Confirmed Erdős's conjecture.

Established existence of integers with many prime sum representations.

Results hold for sufficiently large x and large sets of integers.

Abstract

Let $\mathcal{P}$ denote the set of all primes. In 1950, P. Erd\H{o}s conjectured that if $c$ is an arbitrarily given constant, $x$ is sufficiently large and $a_1,\dots , a_t$ are positive integers with $a_1<a_2<\cdot\cdot\cdot<a_t\leqslant x$ and $t>\log x$, then there exists an integer $n$ so that the number of solutions of $n=p+a_i$ $(p\in \mathcal{P}, 1\le i\le t)$ is greater than $c$. In this note, we confirm this old conjecture of Erd\H{o}s.