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)?
Test
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