Euclid's lemma

In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers:^{[note 1]}

Euclid's lemma — If a prime $p$ divides the product $ab$ of two integers $a$ and $b$ , then $p$ must divide at least one of those integers $a$ or $b$ .

For example, if $p = 19$ , $a = 133$ , $b = 143$ , then $ab = 133 \times 143 = 19019$ , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, $133 = 19 \times 7$ .

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 $p$ is a composite number, its consequent may be either true or false. For example, in the case of $p = 10$ , $a = 4$ , $b = 15$ , composite number 10 divides $ab = 4 \times 15 = 60$ , but 10 divides neither 4 nor 15.

This property is the key in the proof of the fundamental theorem of arithmetic.^{[note 2]} 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.

^ Bajnok 2013, Theorem 14.5
^ Joyner, Kreminski & Turisco 2004, Proposition 1.5.8, p. 25
^ Martin 2012, p. 125

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[1] Bajnok 2013, Theorem 14.5

[2] Joyner, Kreminski & Turisco 2004, Proposition 1.5.8, p. 25

[3] Martin 2012, p. 125

[note 1]

[note 2]