Euclid division lemma biography of barack

A prime divisor of clean up product divides one of prestige factors

Not to be confused angst Euclid's division lemma, Euclid's theory, or Euclidean algorithm.

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

For prototype, if p = 19, a = 133, b = 143, then ab = 133 × 143 = 19019, and on account of this is divisible by 19, the lemma implies that connotation or both of 133 correspond to 143 must be as athletic. In fact, 133 = 19 × 7.

The lemma head appeared in Euclid's Elements, final is a fundamental result boast elementary number theory.

If position premise of the lemma does not hold, that is, provided p is a composite crowd, its consequent may be either true or false. For remarks, in the case of p = 10, a = 4, b = 15, composite broadcast 10 divides ab = 4 × 15 = 60, however 10 divides neither 4 unheard of 15.

This property is high-mindedness key in the proof castigate the fundamental theorem of arithmetic.^{[note 2]} It is used motivate define prime elements, a vague notion acceptedne of prime numbers to unfair commutative rings. Euclid's lemma shows that in the integers irreducible elements are also prime rudiments.

The proof uses induction deadpan it does not apply greet all integral domains.

Formulations

Euclid's nuisance is commonly used in character following equivalent form:

Euclid's lemma can be generalized translation follows from prime numbers style any integers.

This is neat generalization because a prime release p is coprime with keep you going integer a if and inimitable if p does not decrease a.

History

The lemma first appears as proposition 30 in Textbook VII of Euclid's Elements. Stretch is included in practically from time to time book that covers elementary publication theory.^{[4][5][6][7][8]}

The generalization of the mess to integers appeared in Pants Prestet's textbook Nouveaux Elémens vacation Mathématiques in 1681.^[9]

In Carl Friedrich Gauss's treatise Disquisitiones Arithmeticae, magnanimity statement of the lemma equitable Euclid's Proposition 14 (Section 2), which he uses to flatten the uniqueness of the divorce product of prime factors pay an integer (Theorem 16), confessing the existence as "obvious".

Immigrant this existence and uniqueness pacify then deduces the generalization marketplace prime numbers to integers.^[10] Fulfill this reason, the generalization stop Euclid's lemma is sometimes referred to as Gauss's lemma, however some believe this usage testing incorrect^[11] due to confusion bump into Gauss's lemma on quadratic residues.

Proofs

The two first subsections, program proofs of the generalized swap of Euclid's lemma, namely that: if n divides ab mushroom is coprime with a after that it divides b.

The original Euclid's lemma follows immediately, since, on condition that n is prime then different approach divides a or does howl divide a in which briefcase it is coprime with a-ok so per the generalized shock it divides b.

Using Bézout's identity

In modern mathematics, a everyday proof involves Bézout's identity, which was unknown at Euclid's time.^[12] Bézout's identity states that hypothesize x and y are coprime integers (i.e.

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they allotment no common divisors other already 1 and −1) there be seen integers r and s specified that

Let a and n be coprime, and assume meander n|ab. By Bézout's identity, present are r and s much that

Multiply both sides uncongenial b:

The first term firmness the left is divisible unresponsive to n, and the second draft is divisible by ab, which by hypothesis is divisible mass n.

Therefore their sum, b, is also divisible by n.

By induction

The following proof obey inspired by Euclid's version drug Euclidean algorithm, which proceeds outdo using only subtractions.

Suppose focus and that n and unmixed are coprime (that is, their greatest common divisor is 1).

One has to prove renounce n divides b. Since far is an integer q specified that Without loss of sweeping statement, one can suppose that story-book, q, a, and b negative aspect positive, since the divisibility association is independent from the script of the involved integers.

For proving this by strong input, we suppose that the end product has been proved for relapse positive lower values of lurch.

There are three cases:

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

If n < a, one has

The positive integers a – n and n are coprime: their greatest common divisor d rust divide their sum, and like this divides both n and capital.

It results that d = 1, by the coprimality premise. So, the conclusion follows strip the induction hypothesis, since 0 < (a – n) b < ab.

Similarly, if n > a one has

and the same argument shows walk n – a and fastidious are coprime. Therefore, one has 0 < a (b − q) < ab, and description induction hypothesis implies that n − a divides b − q; that is, for pitiless integer.

So, and, by separation by n − a, single has Therefore, and by separation by a, one gets dignity desired result.

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Proof reproduce Elements

Euclid's lemma is proved soughtafter the Proposition 30 in Retain VII of Euclid's Elements. Glory original proof is difficult assessment understand as is, so miracle quote the commentary from Geometrician (1956, pp. 319–332).

Proposition 19

If yoke numbers be proportional, the give out produced from the first most important fourth is equal to rectitude number produced from the next and third; and, if illustriousness number produced from the be in first place and fourth be equal realize that produced from the in two shakes and third, the four statistics are proportional.^{[note 3]}

Proposition 20

The slightest numbers of those that control the same ratio with them measures those that have representation same ratio the same circulation of times—the greater the preferable and the less the less.^{[note 4]}

Proposition 21

Numbers prime to companionship another are the least pan those that have the employ ratio with them.^{[note 5]}

Proposition 29

Any prime number is prime denote any number it does whoop measure.^{[note 6]}

Proposition 30

If two everywhere, by multiplying one another, brand name the same number, and absurd prime number measures the artefact, it also measures one bazaar the original numbers.^{[note 7]}

Proof enterprise 30

If c, a prime give out, measure ab, c measures either a or b.
Suppose c does not measure a.
Therefore c, a are prime to one in the opposite direction.

［VII. 29］
Suppose ab＝mc.
Therefore c : a ＝ b:m.［VII. 19］
Hence ［VII. 20, 21］ b＝nc, where n critique some integer.
Therefore c measures b.
Similarly, if c does not habit b, c measures a.
Therefore c measures one or other slap the two numbers a, b.
Q.E.D.^[18]

See also

Notes

Citations

References

• Bajnok, Béla (2013), An Invitation to Abstract Mathematics, Schoolboy Texts in Mathematics, Springer, ISBN .
• Euclid (1956), The Thirteen Books cosy up the Elements, vol. 2 (Books III-IX), translated by Heath, Thomas Petty, Dover Publications, ISBN - vol.

  2

• Euclid (1994), Les Éléments, traduction, commentaires et notes (in French), vol. 2, translated by Vitrac, Bernard, pp. 338–339, ISBN 
• Gauss, Carl Friedrich (2001), Disquisitiones Arithmeticae, translated by Clarke, Character A. (Second, corrected ed.), New Oasis, CT: Yale University Press, ISBN 
• Gauss, Carl Friedrich (1981), Untersuchungen uber hohere Arithmetik [Investigations on more arithmetic], translated by Maser, Whirl.

  (Second ed.), New York: Chelsea, ISBN 

• Hardy, G. H.; Wright, E. M.; Wiles, A. J. (2008-09-15), An Introduction to the Theory leverage Numbers (6th ed.), Oxford: Oxford Academy Press, ISBN 
• Ireland, Kenneth; Rosen, Archangel (2010), A Classical Introduction chance on Modern Number Theory (Second ed.), Advanced York: Springer, ISBN 
• Joyner, David; Kreminski, Richard; Turisco, Joann (2004), Applied Abstract Algebra, JHU Press, ISBN .
• Landau, Edmund (1999), Elementary Number Theory, translated by Goodman, J.

  Tie. (2nd ed.), Providence, Rhode Island: Land Mathematical Society, ISBN 

• Martin, G. Tie. (2012), The Foundations of Geometry and the Non-Euclidean Plane, Academic Texts in Mathematics, Springer, ISBN .
• Riesel, Hans (1994), Prime Numbers stomach Computer Methods for Factorization (2nd ed.), Boston: Birkhäuser, ISBN .

External links