Polynomials with no Common Factors

Algebra Level 5

Find the number of positive integers n1000,n\leq 1000, such that the polynomials Fn(a,b,c)=a(bc)n+b(ca)n+c(ab)nF_n(a, b, c) = a(b-c)^n+b(c-a)^n+c(a-b)^n and F4(a,b,c)=a(bc)4+b(ca)4+c(ab)4F_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=1n=1, we have F1=0 F_1 = 0 , and hence F4F1 F_4 \mid F_1 .

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