View source for Euclid's lemma

{{Distinguish|Euclid's theorem|Euclidean algorithm}}
In [[number theory]], '''Euclid's lemma''' is a [[lemma (mathematics)|lemma]] that captures a fundamental property of [[prime number]]s, namely:{{refn|group=note|It is also called '''Euclid's first theorem'''<ref>{{Harvnb|Bajnok|2013|loc=Theorem 14.5}}</ref><ref>{{Harvnb|Joyner|Kreminski|Turisco|2004|loc=Proposition 1.5.8, p.&nbsp;25}}</ref> although that name more properly belongs to the [[Solution of triangles|side-angle-side condition]] for showing that [[triangle]]s are [[Congruence (geometry)|congruent]].<ref>{{Harvnb|Martin|2012|p=125}}</ref>}} {{math_theorem|name=Euclid's lemma|math_statement=If a prime {{math|''p''}} divides the product {{math|''ab''}} of two integers {{math|''a''}} and {{math|''b''}}, then {{math|''p''}} must divide at least one of those integers {{math|''a''}} and {{math|''b''}}.}} For example, if {{math|''p'' {{=}} 19}}, {{math|''a'' {{=}} 133}}, {{math|''b'' {{=}} 143}}, then {{math|''ab'' {{=}} 133 &times; 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, {{math| 133 {{=}} 19 × 7}}.

Inherently, if the premise of the lemma does not hold, i.e., {{math|''p''}} is a [[composite number]], its consequent may be either true or false. For example, in the case of {{math|''p'' {{=}} 10}}, {{math|''a'' {{=}} 4}}, {{math|''b'' {{=}} 15}}, composite number 10 divides {{math|''ab'' {{=}} 4 &times; 15 {{=}} 60}}, but 10 divides neither 4 nor 15.

This property is the key in the proof of the [[fundamental theorem of arithmetic]].{{refn|group=note|In general, to show that a [[Integral domain|domain]] is a [[unique factorization domain]], it suffices to prove Euclid's lemma and the [[ascending chain condition on principal ideals]] (ACCP)}} It is used to define [[prime element]]s, a generalization of prime numbers to arbitrary [[commutative ring]]s. Euclid's Lemma shows that in the integers [[irreducible element]]s are also prime elements. The proof uses [[mathematical induction| induction]] so it does not apply to all [[integral domain]]s.

==Formulations==
Let <math>p</math> be a [[prime number]], and assume <math>p</math> divides the product of two integers <math>a</math> and <math>b</math>. (In symbols this is written <math>p \mid ab</math>. Its negation, <math>p</math> does not divide <math>ab</math> is written <math>p \nmid ab</math>.) Then <math>p \mid a</math> or <math>p \mid b</math> (or both). [[Logical equivalence|Equivalent]] statements are:
* If <math>p \nmid a</math> and <math>p \nmid b</math>, then <math>p \nmid ab</math>.
* If <math>p \nmid a</math> and <math>p \mid ab</math>, then <math>p \mid b</math>.

Euclid's lemma can be generalized from prime numbers to any integers:
{{math_theorem|If <math>n \mid ab</math>, and <math>n</math> is [[coprime integers|relatively prime]] to <math>a</math>, then <math>n \mid b</math>.}}
This is a generalization because if <math>n</math> is prime, either
* <math>n \mid a</math> or
* <math>n</math> is relatively prime to <math>a</math>. In this second possibility, <math>n \nmid a</math> so <math>n \mid b</math>.

==History==
The lemma first appears as proposition 30 in Book VII of [[Euclid]]'s ''[[Euclid's Elements|Elements]]''. It is included in practically every book that covers elementary number theory.<ref name="DA1">{{Harvnb|Gauss|2001|p=14}}</ref><ref>{{Harvnb|Hardy|Wright|Wiles|2008|loc=Theorem&nbsp;3}}</ref><ref>{{Harvnb|Ireland|Rosen|2010|loc=Proposition&nbsp;1.1.1}}</ref><ref>{{Harvnb|Landau|Goodman|1999|loc=Theorem&nbsp;15}}</ref><ref>{{Harvnb|Riesel|1994|loc=Theorem&nbsp;A2.1}}</ref>

The generalization of the lemma to integers appeared in [[Jean Prestet]]'s textbook ''Nouveaux Elémens de Mathématiques'' in 1681.<ref>{{Harvnb|Euclid|1994|pp=338–339}}</ref>

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.<ref name="DA2">{{Harvnb|Gauss|2001|loc=Article 19}}</ref> For this reason, the generalization of Euclid's lemma is sometimes referred to as Gauss's lemma, but some believe this usage is incorrect<ref>
{{mathworld
  |title = Euclid's Lemma
  |id = EuclidsLemma
}}</ref> due to confusion with [[Gauss's lemma (number theory)|Gauss's lemma on quadratic residues]].

==Proof==
===Proof using Bézout's lemma===
The usual proof involves another lemma called [[Bézout's identity]].<ref>{{Harvnb|Hardy|Wright|Wiles|2008|loc=§2.10}}</ref> This states that if {{math|''x''}} and {{math|''y''}} are [[coprime integers|relatively prime integers]] (i.e. they share no common divisors other than 1 and -1) there exist integers {{math|''r''}} and {{math|''s''}} such that
:<math>
rx+sy = 1.
</math>
Let {{math|''a''}} and {{math|''n''}} be relatively prime, and assume that {{math|''n''&#124;''ab''}}. By Bézout's identity, there are {{math|''r''}} and {{math|''s''}} making
:<math>
rn+sa = 1.
</math>
Multiply both sides by {{math|''b''}}:
:<math>
rnb+sab = b.
</math>
The first term on the left is divisible by {{math|''n''}}, and the second term is divisible by {{math|''ab''}}, which by hypothesis is divisible by {{math|''n''}}. Therefore their sum, {{math|''b''}}, is also divisible by {{math|''n''}}. This is the generalization of Euclid's lemma mentioned above.

=== Proof of Elements ===
Euclid's lemma is proved at the Proposition 30 in Book VII of ''Elements''. The original proof is difficult to understand as is, so we quote the commentary from {{Harvtxt|Euclid|1956|pp=319-332}}.

;{{Anchor|Proposition 19|}}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.{{refn|group=note|If ''a'' ： ''b''＝''c'' ： ''d'', then ''ad''＝''bc''; and conversely.<ref>{{Harvnb|Euclid|1956|p=319}}</ref>}}
;{{Anchor|Proposition 20|}}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.{{refn|group=note|If ''a'' ： ''b''＝''c'' ： ''d'', and ''a'', ''b'' are the least numbers among those that have the same ratio, then ''c''＝''na'', ''d''＝''nb'', where ''n'' is some integer.<ref>{{Harvnb|Euclid|1956|p=321}}</ref>}}
;{{Anchor|Proposition 21|}}Proposition 21
:Numbers prime to one another are the least of those that have the same ratio with them.{{refn|group=note|If ''a'' ： ''b''＝''c'' ： ''d'', and ''a'', ''b'' are prime to one another, then ''a'', ''b'' are the least numbers among those that have the same ratio.<ref>{{Harvnb|Euclid|1956|p=323}}</ref>}}
;{{Anchor|Proposition 29|}}Proposition 29
:Any prime number is prime to any number it does not measure.{{refn|group=note|If ''a'' is prime and does not measure ''b'', then ''a'', ''b'' are prime to one another.<ref>{{Harvnb|Euclid|1956|p=331}}</ref>}}
;{{Anchor|Proposition 30|}}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.{{refn|group=note|If ''c'', a prime number, measure ''ab'', ''c'' measures either ''a'' or ''b''.<ref>{{Harvnb|Euclid|1956|p=332}}</ref>}}
;Proof
:If ''c'', a prime number, measure ''ab'', ''c'' measures either ''a'' or ''b''.<br />Suppose ''c'' does not measure ''a''.<br />Therefore ''c'', ''a'' are prime to one another. <span style="float: right;">［[[#Proposition 29|VII. 29]]］</span><br />Suppose <span style="text-align: center;">''ab''＝''mc''.</span><br />Therefore <span style="text-align: center;">''c'' ： ''a''＝''b'' ： ''m''.</span> <span style="float: right;">［[[#Proposition 19|VII. 19]]］</span><br />Hence ［[[#Proposition 20|VII. 20]], [[#Proposition 21|21]]］ ''b''＝''nc'', where ''n'' is some integer.<br />Therefore <span style="text-align: center;">''c'' measures ''b''.</span><br />Similarly, if ''c'' does not measure ''b'', ''c'' measures ''a''.<br />Therefore ''c'' measures one or other of the two numbers ''a'', ''b''.<br />[[Q.E.D.]]<ref>{{Harvnb|Euclid|1956|pp=331−332}}</ref>

==See also==
{{Div col}}
*[[Bézout's identity]]
*[[Euclidean algorithm]]
*[[Fundamental theorem of arithmetic]]
*[[Irreducible element]]
*[[Prime element]]
*[[Prime number]]
{{Div col end}}

==Footnotes==
===Notes===
{{reflist|group=note}}

===Citations===
{{reflist|2}}

==References==
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==External links==
*{{MathWorld|title=Euclid's Lemma|id=EuclidsLemma}}

{{DEFAULTSORT:Euclid's Lemma}}
[[Category:Articles containing proofs]]
[[Category:Lemmas]]
[[Category:Theorems about prime numbers]]