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Integer-valued polynomials

Problem 402

Published on Saturday, 17th November 2012, 07:00 pm; Solved by 189

It can be shown that the polynomial n4 + 4n3 + 2n2 + 5n is a multiple of 6 for every integer n. It can also be shown that 6 is the largest integer satisfying this property.

Define M(a, b, c) as the maximum m such that n4 + an3 + bn2 + cn is a multiple of m for all integers n. For example, M(4, 2, 5) = 6.

Also, define S(N) as the sum of M(a, b, c) for all 0 < a, b, cN.

We can verify that S(10) = 1972 and S(10000) = 2024258331114.

Let Fk be the Fibonacci sequence:
F0 = 0, F1 = 1 and
Fk = Fk-1 + Fk-2 for k ≥ 2.

Find the last 9 digits of Σ S(Fk) for 2 ≤ k ≤ 1234567890123.