Euclid's lemma
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In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers:
Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570.
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can be
Euclid's lemma
…of p = 10, a = 4, b = 15, composite number 10 divides ab = 4 × 15 = 60, 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…
In Disquisitiones Arithmeticae (1801) Gauss proved the unique factorization theorem and used it to prove the law of quadratic reciprocity.