Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number

Let $p$ be a prime number and let $f(x)=a_0+a_1x+\ldots+a_nx^n$ with $p\not\mid a_n$ be a polynomial integer coefficients $a_0,\ldots,a_n$ of degree $n.$ Then the Diophantine equation of congruences $f(x)(p)\equiv 0(p)$ has at most $n$ roots (i.e. integers $x$ causing the polynomial to evaluate to $0$ modulo $p$).

Proofs: 1

Proofs: 1 2 3

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  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
  2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998