Insulated primes

TL;DR

This paper introduces the concept of insulated primes based on their maximal prime-free intervals, explores their properties, and presents a new sequence of such primes along with related open problems.

Contribution

It defines the degree of insulation for primes, introduces the sequence of insulated primes, and investigates their properties and relations.

Findings

Identification of a new sequence of insulated primes.

Properties and relations of the degree of insulation.

Open problem related to insulated primes.

Abstract

The degree of insulation of a prime $p$ is defined as the largest interval around it in which no other prime exists. Based on this, the $n$-th prime $p_{n}$ is said to be insulated if and only if its degree of insulation is higher than its neighboring primes. Consequently, a new special sequence emerges given as 7, 13, 23, 37, 53, 67, 89, 103, 113, 131, 139, 157, 173, 181, 193, 211, 233, 277, 293, and so on. This paper presents several properties and intriguing relations concerning degree of insulation and insulated primes. Finally, the reader is left with a captivating open problem.