From a Consequence of Bertrand's Postulate to Hamilton Cycles

TL;DR

This paper explores a graph-theoretic perspective on a number partitioning problem related to Bertrand's postulate, proposing a conjecture about prime-sum cycles and proving it for infinitely many cases.

Contribution

It introduces a new conjecture about prime-sum cycles in integer sequences and proves its validity for infinitely many instances.

Findings

The conjecture holds for infinitely many cases.

Provides new graph-theoretic insights into prime sum problems.

Extends classical number theory results to cycle constructions.

Abstract

A consequence of Bertrand's postulate, proved by L. Greenfield and S. Greenfield in 1998, assures that the set of integers $\{1,2,\cdots, 2n\}$ can be partitioned into pairs so that the sum of each pair is a prime number for any positive integer $n$. Cutting through it from the angle of Graph Theory, this paper provides new insights into the problem. We conjecture a stronger statement that the set of integers $\{1,2,\cdots, 2n\}$ can be rearranged into a cycle so that the sum of any two adjacent integers is a prime number. Our main result is that this conjecture is true for infinitely many cases.