Euclid's lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: 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, . If the premise of the lemma does not hold, i.e., 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. Let formula_1 be a prime number, and assume formula_1 divides the product of two integers formula_3 and formula_4. In symbols, this is written formula_5. Its negation, formula_1 does not divide formula_7, is written formula_8. Then formula_9 or formula_10 (or both). Equivalent statements are: Euclid's lemma can be generalized from prime numbers  to any integers: This is a generalization because if formula_17 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. 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, it is coprime with if and only if it does not  divide . 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 (i.e. they share no common divisors other than 1 and -1) there exist integers and such that Let and be relatively prime, 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 formula_26 and that and are coprime (that is, their greatest common divisor is ). One has to prove that divides . Since formula_27 there is a integer such that formula_28 Without loss of generality, one can suppose that , , , and are positive, since the divisibility relation is independent from the signs of the involved integers. For proving this by strong induction, we suppose that the result has been proved for all positive lower values of . There are three cases: If , coprimality implies  , and divides trivially. If , one has The positive integers and are coprime: their greatest common divisor must divide their sum, and thus divides both and . It results that , by the coprimality hypothesis. So, the conclusion follows from the induction hypothesis, since . Similarly, if one has and the same argument shows that and are coprime. Therefore, one has , and the induction hypothesis implies that divides ; that is, formula_31 for some integer. So, formula_32 and, by dividing by , one has formula_33 Therefore, formula_34 and by dividing by , one gets formula_35 the desired result. 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 .