Polynomials with no Common Factors· Level 5 (335 points)
Find the number of positive integers \(n\leq 1000,\) such that the polynomials \(F_n(a, b, c) = a(b-c)^n+b(c-a)^n+c(a-b)^n\) and \(F_4 (a, b, c) = a(b-c)^4+b(c-a)^4+c(a-b)^4\) have no common non-constant factors.
Details and assumptions
For \(n=1\), we have \( F_1 = 0 \), and hence \( F_4 \mid F_1 \).
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