In number theory, Euclid's lemma (also called Euclid's first theorem) is a lemma that captures one of the fundamental properties of prime numbers. It states that if a prime divides the product of two numbers, it must divide at least one of those numbers. For example since 133 × 143 = 19019 is divisible by 19, one or both of 133 or 143 must be as well. In fact, 19 × 7 = 133. This property is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings (which sometimes differ from irreducible elements).
The lemma is not true for composite numbers. For example, 4 does not divide 6 and 4 does not divide 10, yet 4 does divide their product, 60.
Euclid's lemma is used in certain proofs of the fundamental theorem of arithmetic. Alternatively, the lemma follows directly from the fundamental theorem of arithmetic, which can be proved by elementary means.
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
If p∤a and p∤b, then p∤ab.
If p∤a and p|ab, then p|b.
A generalization is also called Euclid's lemma:
If n|ab, and n is relatively prime to a, then n|b.
This is a generalization because if n is prime, either
- n|a or
- n is relatively prime to a. In this second possibility, so n|b.
This proof depends on the fact that if GCD(a, b) = 1, then LCM(a, b) = ab. Let p be a prime number, and b and c two integers such that p | bc. Suppose p does not divide b. We will show that p must divide c. Let m = LCM(p, b). Since GCD(p, b) = 1, m = pb. Since p | bc and b | bc, bc is a common multiple of p and b. Therefore m | bc, pb | bc, and p | c.
Via Bézout's identity
Another 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, 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.
- Gauss, DA, Art. 14
- Hardy & Wright, Thm. 3
- Ireland & Rosen, prop. 1.1.1
- Landau, Thm. 15
- Riesel, Thm. A2.1
The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
- Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0387962549
- Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8
- Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0198531715
- Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X
- Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea