Euclid's lemma

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

For example, if p = 19, a = 133, b = 143, then ab = 133 × 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 × 7.

If the premise of the lemma does not hold, i.e., 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 × 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.

Formulations[]

Let ${\displaystyle p}$ be a prime number, and assume ${\displaystyle p}$ divides the product of two integers ${\displaystyle a}$ and ${\displaystyle b}$. In symbols, this is written ${\displaystyle p\mid ab}$. Its negation, ${\displaystyle p}$ does not divide ${\displaystyle ab}$, is written ${\displaystyle p\nmid ab}$. Then ${\displaystyle p\mid a}$ or ${\displaystyle p\mid b}$ (or both). Equivalent statements are:

• If ${\displaystyle p\nmid a}$ and ${\displaystyle p\nmid b}$, then ${\displaystyle p\nmid ab}$.
• If ${\displaystyle p\nmid a}$ and ${\displaystyle p\mid ab}$, then ${\displaystyle p\mid b}$.

Euclid's lemma can be generalized from prime numbers to any integers:

This is a generalization because if ${\displaystyle n}$ is prime, either

• ${\displaystyle n\mid a}$ or
• ${\displaystyle n}$ is relatively prime to ${\displaystyle a}$. In this second possibility, ${\displaystyle n\nmid a}$ so ${\displaystyle n\mid b}$.

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.^{[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.

Proof[]

Proof using Bézout's identity[]

In modern mathematics, a common proof involves a result called Bézout's identity, which was unknown at Euclid's time.^[12] Bézout's identity states that if x and y are relatively prime integers (i.e. they share no common divisors other than 1 and -1) there exist integers r and s such that

${\displaystyle rx+sy=1.}$

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

${\displaystyle rn+sa=1.}$

Multiply both sides by b:

${\displaystyle rnb+sab=b.}$

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.

Direct proof[]

The following proof is inspired by Euclid's version of Euclidean algorithm, which proceeds by using only subtractions.

Suppose that ${\displaystyle p\nmid a}$ and ${\displaystyle p\mid ab}$. We want to show that ${\displaystyle p\mid b}$. Since ${\displaystyle p\nmid a}$, there is an n coprime to a (that is, their only common divisors are 1 and –1) such that

${\displaystyle nq=ab.}$

One has to prove that n divides b. For proving this by strong induction, we suppose that this has been proved for all positive lower values of ab.

There are three cases:

If n = a, coprimality implies n = 1, and n divides b trivially.

If n < a, one has

${\displaystyle n(q-b)=(a-n)b.}$

The positive integers a – n and n are coprime: if any prime divides both, then it divides their sum, and thus divides both n and a. This is a contradiction of the coprimality hypothesis. As a consequence of the right hand side , q – b is positive. So, the conclusion follows from the induction hypothesis, since a – n < a.

If n > a, one has

${\displaystyle (n-a)q=a(b-q).}$

Similarly as above, n – a and a are coprime. Since b – q < b, by the induction hypothesis, there is an integer r such that ${\displaystyle b-q=r(n-a).}$ So

${\displaystyle (n-a)q=a(b-q)=ar(n-a),}$

and one gets q = ar, by dividing by n – a. Thus ${\displaystyle ab=nq=anr,}$ and, by division by a, one gets b = nr, which the desired conclusion.

Proof of Elements[]

Euclid's lemma is proved at the Proposition 30 in Book VII of Euclid's Elements. The original proof is difficult to understand as is, so we quote the commentary from Euclid (1956, pp. 319–332).

Proposition 19

If four numbers be proportional, the number produced from the first and fourth is equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers are proportional.^{[note 3]}

Proposition 20

The least numbers of those that have the same ratio with them measures those that have the same ratio the same number of times—the greater the greater and the less the less.^{[note 4]}

Proposition 21

Numbers prime to one another are the least of those that have the same ratio with them.^{[note 5]}

Proposition 29

Any prime number is prime to any number it does not measure.^{[note 6]}

Proposition 30

If two numbers, by multiplying one another, make the same number, and any prime number measures the product, it also measures one of the original numbers.^{[note 7]}

Proof of 30

If c, a prime number, measure ab, c measures either a or b.
Suppose c does not measure a.
Therefore c, a are prime to one another. ［VII. 29］
Suppose ab＝mc.
Therefore c : a ＝ b : m. ［VII. 19］
Hence ［VII. 20, 21］ b＝nc, where n is some integer.
Therefore c measures b.
Similarly, if c does not measure b, c measures a.
Therefore c measures one or other of the two numbers a, b.
Q.E.D.^[18]

See also[]

Footnotes[]

Notes[]

Citations[]

References[]

• Bajnok, Béla (2013), An Invitation to Abstract Mathematics, Undergraduate Texts in Mathematics, Springer, ISBN 978-1-4614-6636-9.
• Euclid (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 |volume= has extra text (help)- vol. 2
• Euclid (1994), Les Éléments, traduction, commentaires et notes (in French), 2, translated by Vitrac, Bernard, pp. 338–339, ISBN 2-13-045568-9
• Gauss, Carl Friedrich (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 (1981), Untersuchungen uber hohere Arithmetik [Investigations on higher arithmetic], translated by Maser, = H. (Second ed.), New York: Chelsea, ISBN 978-0-8284-0191-3
• Hardy, G. H.; 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
• Ireland, Kenneth; Rosen, Michael (2010), A Classical Introduction to Modern Number Theory (Second ed.), New York: Springer, ISBN 978-1-4419-3094-1
• Joyner, David; Kreminski, Richard; Turisco, Joann (2004), Applied Abstract Algebra, JHU Press, ISBN 978-0-8018-7822-0.
• Landau, Edmund; Goodman, J. E. (translator into English) (1999), Elementary Number Theory (2nd ed.), Providence, Rhode Island: American Mathematical Society, ISBN 978-0-821-82004-9
• Martin, G. E. (2012), The Foundations of Geometry and the Non-Euclidean Plane, Undergraduate Texts in Mathematics, Springer, ISBN 978-1-4612-5725-7.
• Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (2nd ed.), Boston: Birkhäuser, ISBN 978-0-8176-3743-9.

External links[]

• Weisstein, Eric W. "Euclid's Lemma". MathWorld.