Problem 269

Problem 269

Polynomials with at least one integer root

A root or zero of a polynomial P(x) is a solution to the equation P(x) = 0.
Define Pn as the polynomial whose coefficients are the digits of n.
For example, P5703(x) = 5x3 + 7x2 + 3.

We can see that:

  • Pn(0) is the last digit of n,
  • Pn(1) is the sum of the digits of n,
  • Pn(10) is n itself.

Define Z(k) as the number of positive integers, n, not exceeding k for which the polynomial Pn has at least one integer root.

It can be verified that Z(100 000) is 14696.

What is Z(1016)?


多项式P(x)的根或零点,是方程P(x) = 0的解。
例如,P5703(x) = 5x3 + 7x2 + 3。


  • Pn(0)是n的个位数字。
  • Pn(1)是n的各位数字和。
  • Pn(10)是n本身。


可以验证Z(100 000)为14696。