History[edit]

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.[4][5][6][7][8]


The generalization of the lemma to integers appeared in Jean Prestet's textbook Nouveaux Elémens de Mathématiques in 1681.[9]


In Carl Friedrich Gauss's treatise Disquisitiones Arithmeticae, the statement of the lemma is Euclid's Proposition 14 (Section 2), which he uses to prove the uniqueness of the decomposition product of prime factors of an integer (Theorem 16), admitting the existence as "obvious". From this existence and uniqueness he then deduces the generalization of prime numbers to integers.[10] For this reason, the generalization of Euclid's lemma is sometimes referred to as Gauss's lemma, but some believe this usage is incorrect[11] due to confusion with Gauss's lemma on quadratic residues.

Bajnok, Béla (2013), , Undergraduate Texts in Mathematics, Springer, ISBN 978-1-4614-6636-9.

An Invitation to Abstract Mathematics

(1956), The Thirteen Books of the Elements, vol. 2 (Books III-IX), translated by Heath, Thomas Little, Dover Publications, ISBN 978-0-486-60089-5- vol. 2

Euclid

(1994), Les Éléments, traduction, commentaires et notes (in French), vol. 2, translated by Vitrac, Bernard, pp. 338–339, ISBN 2-13-045568-9

Euclid

(2001), Disquisitiones Arithmeticae, translated by Clarke, Arthur A. (Second, corrected ed.), New Haven, CT: Yale University Press, ISBN 978-0-300-09473-2

Gauss, Carl Friedrich

Gauss, Carl Friedrich (1981), Untersuchungen uber hohere Arithmetik [Investigations on higher arithmetic], translated by Maser, H. (Second ed.), New York: Chelsea,  978-0-8284-0191-3

ISBN

; Wright, E. M.; Wiles, A. J. (2008-09-15), An Introduction to the Theory of Numbers (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5

Hardy, G. H.

Ireland, Kenneth; Rosen, Michael (2010), A Classical Introduction to Modern Number Theory (Second ed.), New York: , ISBN 978-1-4419-3094-1

Springer

Joyner, David; Kreminski, Richard; Turisco, Joann (2004), , JHU Press, ISBN 978-0-8018-7822-0.

Applied Abstract Algebra

(1999), Elementary Number Theory, translated by Goodman, J. E. (2nd ed.), Providence, Rhode Island: American Mathematical Society, ISBN 978-0-821-82004-9

Landau, Edmund

Martin, G. E. (2012), , Undergraduate Texts in Mathematics, Springer, ISBN 978-1-4612-5725-7.

The Foundations of Geometry and the Non-Euclidean Plane

(1994), Prime Numbers and Computer Methods for Factorization (2nd ed.), Boston: Birkhäuser, ISBN 978-0-8176-3743-9.

Riesel, Hans

"Euclid's Lemma". MathWorld.

Weisstein, Eric W.