Euclid's lemma


In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers:
For example, if,,, then, and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact,.
The lemma first appeared in Euclid's Elements, and is a fundamental result in elementary number theory.
If the premise of the lemma does not hold, that is, if is a composite number, its consequent may be either true or false. For example, in the case of,,, composite number 10 divides, but 10 divides neither 4 nor 15.
This property is the key in the proof of the fundamental theorem of arithmetic. It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings. Euclid's lemma shows that in the integers irreducible elements are also prime elements. The proof uses induction so it does not apply to all integral domains.

Formulations

Euclid's lemma is commonly used in the following equivalent form:
Euclid's lemma can be generalized as follows from prime numbers to any integers.
This is a generalization because a prime number is coprime with an integer if and only if does not divide.

History

The lemma first appears as proposition 30 in Book VII of Euclid's Elements. It is included in practically every book that covers elementary number theory.
The generalization of the lemma to integers appeared in Jean Prestet's textbook Nouveaux Elémens de Mathématiques in 1681.
In Carl Friedrich Gauss's treatise Disquisitiones Arithmeticae, the statement of the lemma is Euclid's Proposition 14, which he uses to prove the uniqueness of the decomposition product of prime factors of an integer, admitting the existence as "obvious". From this existence and uniqueness he then deduces the generalization of prime numbers to integers. For this reason, the generalization of Euclid's lemma is sometimes referred to as Gauss's lemma, but some believe this usage is incorrect due to confusion with Gauss's lemma on quadratic residues.

Proofs

The two first subsections, are proofs of the generalized version of Euclid's lemma, namely that: if divides and is coprime with then it divides.
The original Euclid's lemma follows immediately, since, if is prime then it divides or does not divide in which case it is coprime with so per the generalized version it divides.

Using Bézout's identity

In modern mathematics, a common proof involves Bézout's identity, which was unknown at Euclid's time. Bézout's identity states that if and are coprime integers there exist integers and such that
Let and be coprime, and assume that. By Bézout's identity, there are and such that
Multiply both sides by :
The first term on the left is divisible by, and the second term is divisible by, which by hypothesis is divisible by. Therefore their sum,, is also divisible by.

By induction

The following proof is inspired by Euclid's version of Euclidean algorithm, which proceeds by using only subtractions.
Suppose that and that and are coprime. One has to prove that divides. Since there exists an integer such that
Without loss of generality, one can suppose that,,, and are positive, since the divisibility relation is independent of the signs of the involved integers.
To prove the theorem by strong induction, we suppose that it has been proved for all smaller values of. There are three cases:
  1. If, coprimality implies, and divides trivially.
  2. :
  3. If, then subtracting from both sides gives



Thus, divides. Since we assumed that and are coprime, it follows that and must be coprime.

The conclusion therefore follows by induction hypothesis, since.
  1. :
  2. If then subtracting from both sides gives



Thus, divides. Since, and are coprime, and since, then the induction hypothesis implies that divides ; that is, for some integer. So, and, by dividing by, one has Therefore, and by dividing by, one gets the desired conclusion.

Proof of ''Elements''

Euclid's lemma is proved at the Proposition 30 in Book VII of Euclid's Elements. The original proof is difficult to understand as is, so we quote the commentary from.
;Proposition 19
;Proposition 20
;Proposition 21
;Proposition 29
;Proposition 30
;Proof of 30