# Prime Divisors Positive Difference

**Number Theory**Level 5

For a composite positive integer \(x\), denote by \(pd(x)\) the smallest positive difference between any two prime divisors of \(x.\) Find the smallest possible value of \(pd(x)\) for composite \(x\) of the form \(x=n^{100}+n^{99}+\cdots+n+1,\) where \(n\) is a positive integer.

**Details and assumptions**

You may refer to a list of primes.

For example, since \( 429 = 3 \times 11 \times 13\), \( pd(429) = | 13 - 11 | = 2 \).