Primes and Functions

Let $$p$$ be a prime number and let $$f(x)$$ be a polynomial of degree $$d$$ with integer coefficients such that:

(i) $$f(0) = 0, f(1) = 1$$

(ii) for every positive integer $$n$$, the remainder upon division of $$f(n)$$ by $$p$$ is either 0 or 1.

Prove the $$d \geq p - 1$$.

Note by Sharky Kesa
4 years, 2 months ago

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I am going to prove a weaker result which does not use condition (i).

Consider the polynomial $$g(x)=f(x)(f(x)-1)$$ of degree $$2d$$. According to (ii) $$g(x) \mod(p) = 0$$ at each of $$x=0,1,2,3,\ldots, p-1$$. Viewing $$g(x)$$ as a polynomial over the field $$\mathbb{F}_p$$, by fundamental theorem of algebra, we immediately conclude that $$2d \geq p$$, i.e. $$d\geq \frac{p}{2}$$.

- 4 years, 1 month ago