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)?

These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK