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Problem 402


Problem 402


Integer-valued polynomials

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, c ≤ N.

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.


值为整数的多项式

可以验证,对于任意整数n,n4 + 4n3 + 2n2 + 5n是6的倍数,而6是最大的满足这一性质的整数。

若对于任意整数n,n4 + an3 + bn2 + cn是整数m的倍数,记M(a, b, c)为m的最大值。例如,M(4, 2, 5) = 6。

记S(N)是所有满足0 < a, b, c ≤ N的M(a, b, c)的和。

我们可以验证S(10) = 1972,以及S(10000) = 2024258331114。

记Fk为斐波那契数列:
F0 = 0,F1 = 1 并且对于任意k ≥ 2
Fk = Fk-1 + Fk-2

对于2 ≤ k ≤ 1234567890123,求Σ S(Fk)的最后9位数字。