# Symmetric primes revisited

## TL;DR

This paper improves a 1996 theorem on symmetric prime pairs, establishing a likely optimal upper bound and proving the infinitude of such pairs, including strings of consecutive primes all forming symmetric pairs.

## Contribution

It refines the upper bound on symmetric prime pairs and proves the existence of infinitely many such pairs, including consecutive prime strings.

## Key findings

- Improved the upper bound on symmetric prime pairs.
- Proved infinitely many symmetric prime pairs exist.
- Established existence of consecutive prime strings with symmetric pairs.

## Abstract

A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a symmetric pair. In the present paper, that theorem is improved to what is likely to be the best possible result. We also establish that there exist infinitely many symmetric pairs of primes. In fact, we show that for every integer m at least 2 there is a string of m consecutive primes, any two of which form a symmetric pair.