Insulated primes

Abhimanyu Kumar, Anuraag Saxena

Published 2020-11-28Version 1

The degree of insulation $D(p)$ of a prime $p$ is defined as the largest interval around the prime $p$ in which no other prime is present. 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 surrounding primes. Thus, a new special sequence emerges which is 7, 13, 23, 37, 53, 67, 89, 103, 113, 131, 139, 157, 173, 181, 193, 211, 233, 277, 293, and so on. Therefore, this paper thoroughly investigates its properties and explores the connection of $D(p)$ with the gap between consecutive primes. Heuristically, it is shown that $n$-th insulated prime grows almost linearly, which is a property in contrast to the effort to obtain a general formula for $n$-th prime. Finally, the paper leaves the readers with a captivating open problem.