Euclid's Lemma
Revision as of 21:31, 13 May 2008 by Boy Soprano II (talk | contribs) (deleted fallacious proof; mentioned converse in proof; edited a little)
Euclid's Lemma is a result in number theory attributed to Euclid. It states that:
A positive integer is a prime number if and only if implies that or , for all integers and .
Proof of Euclid's Lemma
Without loss of generality, suppose (otherwise we are done). By Bezout's Lemma, there exist integers such that such that . Hence and . Since and (by hypothesis), , as desired.
On the other hand, if is not prime, then it must be composite, i.e., , for integers both greater than 1. Then and . Thus the lemma's converse holds as well.