Challenge Question:
Suppose I have a polynomial with positive integral coefficients. You give me an integer and I give you . You then give me an integer and I give you . Your objective is to determine .
How would you do so?
It's probably easiest to see this by looking at an example. Suppose that you think of a polynomial (with positive integral coefficients). You don't tell me what it is, so all I know is that it is some expression of the form . I ask you for p(1) and you tell me that p(1)=9. That tells me that . In particular, each of the coefficients must lie between 0 and 9.
I then ask you for p(10) and you tell me that p(10) = 1243. I then express that number in base 10. (Of course, I have made this example easy, because 1243 is already in base 10.) So I know that . But there is a unique way to express a number in base 10, with all the digits lying between 0 and 9, so I can immediately read off that n=3 and , , and .
At that point, I can astonish you by revealing that your polynomial was .