# 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 $$429 = 3 \times 11 \times 13$$, $$pd(429) = | 13 - 11 | = 2$$.

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