yaworsw/euler-manager

---
:id: 402
:name: Integer-valued polynomials
:url: https://projecteuler.net/problem=402
:content: "It can be shown that the polynomial <var>n</var><sup>4</sup> + 4<var>n</var><sup>3</sup>
  + 2<var>n</var><sup>2</sup> + 5<var>n</var> is a multiple of 6 for every integer
  <var>n</var>. It can also be shown that 6 is the largest integer satisfying this
  property.\n\nDefine M(<var>a</var>, <var>b</var>, <var>c</var>) as the maximum <var>m</var>
  such that <var>n</var><sup>4</sup> + <var>a</var><var>n</var><sup>3</sup> + <var>b</var><var>n</var><sup>2</sup>
  + <var>c</var><var>n</var> is a multiple of <var>m</var> for all integers <var>n</var>.
  For example, M(4, 2, 5) = 6.\n\nAlso, define S(<var>N</var>) as the sum of M(<var>a</var>,
  <var>b</var>, <var>c</var>) for all 0 \\< <var>a</var>, <var>b</var>, <var>c</var>
  ≤ <var>N</var>.\n\nWe can verify that S(10) = 1972 and S(10000) = 2024258331114.\n\nLet
  F<sub><var>k</var></sub> be the Fibonacci sequence:  \nF<sub>0</sub> = 0, F<sub>1</sub>
  = 1 and  \nF<sub><var>k</var></sub> = F<sub><var>k</var>-1</sub> + F<sub><var>k</var>-2</sub>
  for <var>k</var> ≥ 2.\n\nFind the last 9 digits of Σ S(F<sub><var>k</var></sub>)
  for 2 ≤ <var>k</var> ≤ 1234567890123.\n\n"