Question 30
Let $n \geq 4$ be an integer. Consider a regular $2n$-gon in which an integer is assigned to each vertex, which we call the value of that vertex. If four distinct vertices of this $2n$-gon form a rectangle, we call the sum of the values of these vertices a rectangle sum.
Determine for which integers $m$ (not necessarily positive) we can assign the numbers $m+1, m+2, \ldots, m+2n$ as values to the vertices (in some order) such that every rectangle sum is a prime number. (Prime numbers are positive by definition.)
Determine for which integers $m$ (not necessarily positive) we can assign the numbers $m+1, m+2, \ldots, m+2n$ as values to the vertices (in some order) such that every rectangle sum is a prime number. (Prime numbers are positive by definition.)