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Polynomials with no Common Factors

Algebra Level 5 400 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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