Euclid's lemma

Not to be confused with Euclid's theorem or Euclid's algorithm.

In number theory, Euclid's lemma (also called Euclid's first theorem) is a lemma that captures a fundamental property of prime numbers, namely: If a prime divides the product of two numbers, it must divide at least one of those numbers. For example, 133 × 143 = 19019. Since 19019 is divisible by 19, one or both of 133 or 143 must be as well. In fact, 133 ÷ 19 = 7.

This property is the key[1] 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.

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.

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 pab.)
Then p|a or p|b (or both).

Equivalent statements are

If pa and pb, then pab.
If pa 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

1. n|a or
2. n is relatively prime to a. In this second possibility, na so n|b.

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.[2][3][4][5][6]

Proof via Bézout's identity

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

$rx+sy = 1.$

Let a and n be relatively prime, and assume that n | ab. By Bézout, there are r and s making

$rn+sa = 1.$

Multiply both sides by b:

$rnb+sab = b.$

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.

Notes

1. ^ In general, to show that a domain is a unique factorization domain, it suffices to prove Euclid's lemma and ACCP.
2. ^ Gauss, DA, Art. 14
3. ^ Hardy & Wright, Thm. 3
4. ^ Ireland & Rosen, prop. 1.1.1
5. ^ Landau, Thm. 15
6. ^ Riesel, Thm. A2.1

References

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
• 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
• Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 0-8176-3743-5