Soljan1806

$x n + 1 - (x + 1) n = 2001$

This is a polynomial with constant term -2002. By the rational roots theorem any solution for x must be a divisor of 2002. Also any solution has to be a solution modulo any base we choose. The only divisors of 2002 which are solutions mod 3,5, and 13 are 13 and 2002. Both give solutions but $n$ for 2002 is 0. We can use calculus to show that any solution with $x = 13$ has $n < 35$ . The only value of $n$ which works is $n = 2$ .

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