Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$.
To determine the minimum number of elements in \( S \) such that there exists a function \( f: \mathbb{N} \rightarrow S \) with the property that if \( x \) and \( y \) are positive integers whose difference is a prime number, then \( f(x) \neq f(y) \), we will proceed as follows:

1. **Construct a function \( f \) with \( |S| = 4 \):**
   Define the function \( f \) as follows:
   \[
   f(4k) = 4, \quad f(4k+1) = 1, \quad f(4k+2) = 2, \quad f(4k+3) = 3
   \]
   for \( k \in \mathbb{N} \).

2. **Verify that this construction works:**
   We need to check that for any \( x \) and \( y \) such that \( |x - y| \) is a prime number, \( f(x) \neq f(y) \).

   - If \( x \equiv 0 \pmod{4} \), then \( f(x) = 4 \). The possible values of \( y \) such that \( |x - y| \) is a prime number are \( y \equiv 1, 2, 3 \pmod{4} \). For these values, \( f(y) = 1, 2, 3 \) respectively, which are all different from 4.
   - If \( x \equiv 1 \pmod{4} \), then \( f(x) = 1 \). The possible values of \( y \) such that \( |x - y| \) is a prime number are \( y \equiv 0, 2, 3 \pmod{4} \). For these values, \( f(y) = 4, 2, 3 \) respectively, which are all different from 1.
   - If \( x \equiv 2 \pmod{4} \), then \( f(x) = 2 \). The possible values of \( y \) such that \( |x - y| \) is a prime number are \( y \equiv 0, 1, 3 \pmod{4} \). For these values, \( f(y) = 4, 1, 3 \) respectively, which are all different from 2.
   - If \( x \equiv 3 \pmod{4} \), then \( f(x) = 3 \). The possible values of \( y \) such that \( |x - y| \) is a prime number are \( y \equiv 0, 1, 2 \pmod{4} \). For these values, \( f(y) = 4, 1, 2 \) respectively, which are all different from 3.

   Thus, the function \( f \) defined above satisfies the required condition.

3. **Show that \( |S| < 4 \) does not work:**
   Suppose \( |S| < 4 \). Then \( |S| \) can be at most 3. Consider the values \( f(1), f(3), f(6), f(8) \). The differences between these values are:
   \[
   |1 - 3| = 2, \quad |3 - 6| = 3, \quad |6 - 8| = 2, \quad |1 - 6| = 5, \quad |1 - 8| = 7, \quad |3 - 8| = 5
   \]
   All these differences are prime numbers. Therefore, \( f(1), f(3), f(6), f(8) \) must all be different. However, if \( |S| < 4 \), it is impossible to assign different values to these four numbers, leading to a contradiction.

Therefore, the minimum number of elements in \( S \) is 4.

The final answer is \( \boxed{4} \).