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。