n⁹ - 6n⁷ + 9n⁵ - 4n³ is divisible by 8640.

Proof is by induction :

for n = 1 , P(1) = 0 divisible by 8640 ;true

Assume it is true for n = k :

k⁹ - 6k⁷ + 9k⁵ - 4k³ = 8640p for some integer p.

 P(k + 1) = (k+1)⁹ - 6(k+1)⁷ + 9(k+1)⁵ - 4(k+1)³

             = k²(k+3)(k-1)(k+2)²(k+1)³

But P(k) =  k⁹ - 6k⁷ + 9k⁵ - 4k³ = k³(k-2)(k+2)(k-1)²(k+1)² =8640p

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