Lagrange's theorem (number theory)

In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, it states that if p is a prime number, ${\displaystyle x\in \mathbb {Z} /p\mathbb {Z} }$, and ${\displaystyle \textstyle f(x)\in \mathbb {Z} [x]}$ is a polynomial with integer coefficients, then either:

• every coefficient of f(x) is divisible by p, or
• f(x) ≡ 0 (mod p) has at most deg f(x) solutions

where deg f(x) is the degree of f(x). If the modulus is not prime, then it is possible for there to be more than deg f(x) solutions.