Lemmas in number theory

In number theory, a lemma is a proven statement or proposition that is used as a stepping stone to prove a larger theorem. The term "lemma" comes from the Greek word "lemma," which means "that which is taken" or "premise." Lemmas can be thought of as auxiliary results that help in the development of more complex arguments or proofs.

Euclid's lemma is a fundamental statement in number theory that relates to the properties of prime numbers and divisibility. It states: **If a prime number \( p \) divides the product of two integers \( a \) and \( b \) (i.e., \( p \mid (a \cdot b) \)), then \( p \) must divide at least one of those integers \( a \) or \( b \) (i.e.

The Lifting-the-exponent lemma (LTE) is a mathematical result in number theory that provides conditions under which the highest power of a prime \( p \) that divides certain expressions can be easily determined. It simplifies the computation of \( v_p(a^n - b^n) \) and related expressions, where \( v_p(x) \) denotes the p-adic valuation, which gives the exponent of the highest power of \( p \) that divides \( x \).