Problems 66 and 67 on sums of residue classes and primes of Andr\'{a}s S\'{a}rk\"{o}zy's collection of unsolved problems

TL;DR

This paper addresses two open problems posed by Sárközy, providing an optimal result concerning sumsets of sparse prime subsets, advancing understanding in additive number theory.

Contribution

It offers a new optimal result on sumsets of sparse prime subsets, solving two of Sárközy's longstanding problems from 2001.

Findings

Proved an optimal result on sumsets of sparse prime subsets

Resolved two of Sárközy's unsolved problems from 2001

Enhanced understanding of additive properties of primes

Abstract

In this note, we discuss two problems of S\'{a}rk\"{o}zy (2001). In particular, we prove an optimal result on sumsets of sparse subset of primes.