From a Consequence of Bertrand's Postulate to Hamilton Cycles

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.