In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely:
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For example, if p = 19, a = 133, b = 143, then ab = 133 × 143 = 19019, and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, 133 = 19 × 7.
The lemma is not true for composite numbers. For example, in the case of p = 10, a = 4, b = 15, composite number 10 divides ab = 4 × 15 = 60, 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
Let p be a prime number, and assume p divides the product of two integers a and b. (In symbols this is written p|ab. Its negation, p does not divide ab is written p∤ab.) Then p|a or p|b (or both). Equivalent statements are:
Euclid's lemma can be generalized from prime numbers to any integers:
This is a generalization because if n is prime, either
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 (Section 2), which he uses to prove the uniqueness of the decomposition product of prime factors of an integer (Theorem 16), 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.
Proof using Bézout's lemma
The usual proof involves another lemma called Bézout's identity. This states that if x and y are relatively prime integers (i.e. they share no common divisors other than 1) there exist integers r and s such that
Let a and n be relatively prime, and assume that n|ab. By Bézout's identity, there are r and s making
Multiply both sides by b:
The first term on the left is divisible by n, and the second term is divisible by ab, which by hypothesis is divisible by n. Therefore their sum, b, is also divisible by n. This is the generalization of Euclid's lemma mentioned above.
Proof of Elements
Euclid's lemma is proved at the Proposition 30 in Book VII of Elements. The original proof is difficult to understand as is, so we quote the commentary from Euclid & Heath (1956, pp. 319-332).
Suppose c does not measure a.
Therefore c, a are prime to one another. ［VII. 29］
Suppose ab＝mc.
Therefore c ： a＝b ： m. ［VII. 19］
Hence ［VII. 20, 21］ b＝nc, where n is some integer.
Therefore c measures b.
Similarly, if c does not measure b, c measures a.
Therefore c measures one or other of the two numbers a, b.
Q.E.D.