View source for Fundamental theorem of arithmetic

{{short description|Integers have unique prime factorizations}}
{{about|the unique factorization of integers into prime numbers|the theorem about roots of a polynomial|Fundamental theorem of algebra|other fundamental theorems|List of theorems called fundamental}}
[[File:Disqvisitiones-800.jpg|thumb|In ''[[Disquisitiones Arithmeticae]]'' (1801) [[Carl Friedrich Gauss|Gauss]] proved the unique factorization theorem <ref name="Gauss1801.loc=16">{{Harvtxt|Gauss|1986|loc=Art. 16}}</ref> and used it to prove the [[law of quadratic reciprocity]].<ref>{{Harvtxt|Gauss|1986|loc=Art. 131}}</ref>]]
In [[mathematics]], the '''fundamental theorem of arithmetic''', also called the '''unique factorization theorem''' and '''prime factorization theorem''', states that every [[integer]] greater than 1 can be represented uniquely as a product of [[prime number]]s, [[up to]] the order of the factors.<ref>{{harvtxt|Long|1972|p=44}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=53}}</ref><ref>{{Harvtxt|Hardy|Wright|2008|loc=Thm 2}}</ref> For example,
:<math>
1200 = 2^4 \cdot 3^1 \cdot 5^2 = (2 \cdot 2 \cdot 2 \cdot 2) \cdot 3 \cdot (5 \cdot 5) = 5 \cdot 2 \cdot 5 \cdot 2 \cdot 3 \cdot 2 \cdot 2 = \ldots
</math>

The theorem says two things about this example: first, that 1200 {{em|can}} be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product.

The requirement that the factors be prime is necessary: factorizations containing [[composite number]]s may not be unique 
(for example, <math>12 = 2 \cdot 6 = 3 \cdot 4</math>).

This theorem is one of the main [[Prime number#Primality of one|reasons why 1 is not considered a prime number]]: if 1 were prime, then factorization into primes would not be unique; for example, <math>2 = 2 \cdot 1 = 2 \cdot 1 \cdot 1 = \ldots</math>

The theorem generalizes to other [[algebraic structure]]s that are called [[unique factorization domain]]s and include [[principal ideal domain]]s, [[Euclidean domain]]s, and [[polynomial ring]]s over a [[field (mathematics)|field]]. However, the theorem does not hold for [[algebraic integer]]s.<ref>In a [[ring of algebraic integers]], the factorization into prime elements may be non unique, but one can recover a unique factorization if one factors into [[ideal (ring theory)|ideal]]s.</ref> This failure of unique factorization is one of the reasons for the difficulty of the proof of [[Fermat's Last Theorem]]. The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between [[Pierre de Fermat|Fermat's]] statement and [[Wiles's proof of Fermat's Last Theorem|Wiles's proof]].

==History==

The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of [[Euclid]]'s ''[[Euclid's Elements|Elements]]''.
{{Quotation|
If two numbers by multiplying one another make some
number, and any prime number measure the product, it will
also measure one of the original numbers.
|Euclid|[[#CITEREFEuclidHeath1956|Elements Book VII]], Proposition 30}}

(In modern terminology: if a  prime ''p'' divides the product ''ab'', then ''p'' divides either ''a'' or ''b'' or both.) Proposition 30 is referred to as [[Euclid's lemma]], and it is the key in the proof of the fundamental theorem of arithmetic.

{{Quotation|
Any composite number is measured by some prime number.
|Euclid|[[#CITEREFEuclidHeath1956|Elements Book VII]], Proposition 31}}

(In modern terminology: every integer greater than one is divided evenly by some prime number.) Proposition 31 is proved directly by [[Proof by infinite descent|infinite descent]].

{{Quotation|
Any number either is prime or is measured by some prime number.
|Euclid|[[#CITEREFEuclidHeath1956|Elements Book VII]], Proposition 32}}

Proposition 32 is derived from proposition 31, and proves that the decomposition is possible.

{{Quotation|
If a number be the least that is measured by prime numbers, it will not be measured by any
other prime number except those originally measuring it.
|Euclid|[[#CITEREFEuclidHeath1956|Elements Book IX]], Proposition 14}}

(In modern terminology: a [[least common multiple]] of several prime numbers is not a multiple of any other prime number.) Book IX, proposition 14 is derived from Book VII, proposition 30, and proves partially that the decomposition is unique – a point critically noted by [[André Weil]].<ref>{{Harvtxt|Weil|2007|p=5}}: "Even in Euclid, we fail to find a general statement about the uniqueness of the factorization of an integer into primes; surely he may have been aware of it, but all he has is a statement (Eucl.IX.I4) about the l.c.m. of any number of given primes."</ref> Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case.

While [[Euclid]] took the first step on the way to the existence of prime factorization, [[Kamāl al-Dīn al-Fārisī]] took the final step<ref>{{Cite journal |last=A. Goksel Agargun and E. Mehmet Özkan |title=A Historical Survey of the Fundamental Theorem of Arithmetic |url=https://core.ac.uk/download/pdf/82721726.pdf |journal=Historia Mathematica |pages=209|quote=One could say that Euclid takes the first step on the way to the existence of prime factorization, and al-Farisi takes the final step by actually proving the existence of a finite prime factorization in his first proposition.}}</ref> and stated for the first time the fundamental theorem of arithmetic.<ref>{{Cite book|url=https://books.google.com/books?id=7veIAgAAQBAJ&q=fundamental+theorem+of+arithmetic+discovered+al-farisi&pg=PA385|title=Encyclopedia of the History of Arabic Science|last=Rashed|first=Roshdi|date=2002-09-11|publisher=Routledge|isbn=9781134977246|language=en|page=385|quote=The famous physicist and mathematician Kamal al-Din al-Farisi compiled a paper in which he set out deliberately to prove the theorem of Ibn Qurra in an algebraic way. This forced him to an understanding of the first arithmetical functions and to a full preparation which led him to state for the first time the fundamental theorem of arithmetic.}}</ref>

Article 16 of [[Carl Friedrich Gauss|Gauss]]'s ''[[Disquisitiones Arithmeticae]]'' is an early modern statement and proof employing [[modular arithmetic]].<ref name="Gauss1801.loc=16" />

==Applications==
===Canonical representation of a positive integer=== <!-- Redirect [[Canonical form]] links here -->

Every positive integer {{math|''n'' > 1}} can be represented in exactly one way as a product of prime powers
:<math>
n = p_1^{n_1}p_2^{n_2} \cdots p_k^{n_k}
= \prod_{i=1}^{k} p_i^{n_i},
</math>
where {{math|''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>k</sub>}} are primes and the {{math|''n''<sub>''i''</sub>}} are positive integers. This representation is commonly extended to all positive integers, including 1, by the convention that the [[empty product]] is equal to 1 (the empty product corresponds to {{math|1=''k'' = 0}}).

This representation is called the '''canonical representation'''<ref>{{harvtxt|Long|1972|p=45}}</ref> of {{math|''n''}}, or the '''standard form'''<ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=55}}</ref><ref>{{Harvtxt|Hardy|Wright|2008|loc=§ 1.2}}</ref> of ''n''.  For example,

:999 = 3<sup>3</sup>×37,
:1000 = 2<sup>3</sup>×5<sup>3</sup>,
:1001 = 7×11×13.

Factors {{math|1=''p''<sup>0</sup> = 1}} may be inserted without changing the value of {{math|''n''}} (for example, {{math|1=1000 = 2<sup>3</sup>×3<sup>0</sup>×5<sup>3</sup>}}).  In fact, any positive integer can be uniquely represented as an [[infinite product]] taken over all the positive prime numbers, as
:<math>n=2^{n_1}3^{n_2}5^{n_3}7^{n_4}\cdots=\prod_{i=1}^\infty p_i^{n_i},</math>
where a finite number of the {{math|''n''<sub>''i''</sub>}} are positive integers, and the others are zero. 

Allowing negative exponents provides a canonical form for positive [[rational number]]s.

===Arithmetic operations===
The canonical representations of the product, [[greatest common divisor]] (GCD), and [[least common multiple]] (LCM) of two numbers ''a'' and ''b'' can be expressed simply in terms of the canonical representations of ''a'' and ''b'' themselves:
:<math>\begin{alignat}{2}
 a\cdot b & = 2^{a_1+b_1}3^{a_2+b_2}5^{a_3+b_3}7^{a_4+b_4}\cdots
   && = \prod p_i^{a_i+b_i},\\
 \gcd(a,b) & = 2^{\min(a_1,b_1)}3^{\min(a_2,b_2)}5^{\min(a_3,b_3)}7^{\min(a_4,b_4)}\cdots
   && = \prod p_i^{\min(a_i,b_i)},\\
 \operatorname{lcm}(a,b) & = 2^{\max(a_1,b_1)}3^{\max(a_2,b_2)}5^{\max(a_3,b_3)}7^{\max(a_4,b_4)}\cdots
   && = \prod p_i^{\max(a_i,b_i)}.
 \end{alignat}</math>

However, [[integer factorization]], especially of large numbers, is much more difficult than computing products, GCDs, or LCMs.  So these formulas have limited use in practice.

===Arithmetic functions===
{{Main article|Arithmetic function}}

Many arithmetic functions are defined using the canonical representation. In particular, the values of [[additive function|additive]] and [[multiplicative function|multiplicative]] functions are determined by their values on the powers of prime numbers.

==Proof==
The proof uses [[Euclid's lemma]] (''Elements'' VII, 30): If a prime [[Divisor|divides]] the product of two integers, then it must divide at least one of these integers.

===Existence===
It must be shown that every integer greater than {{math|1}} is either prime or a product of primes.  First, {{math|2}} is prime.  Then, by [[strong induction]], assume this is true for all numbers greater than {{math|1}} and less than {{math|''n''}}. If {{math|''n''}} is prime, there is nothing more to prove. Otherwise, there are integers {{math|''a''}} and {{math|''b''}},  where {{math|1=''n'' = ''a b''}}, and {{math|1 < ''a'' ≤ ''b'' < ''n''}}.  By the induction hypothesis, {{math|1=''a'' = ''p''<sub>1</sub> ''p''<sub>2</sub> ⋅⋅⋅ ''p''<sub>''j''</sub>}} and {{math|1=''b'' = ''q''<sub>1</sub> ''q''<sub>2</sub> ⋅⋅⋅ ''q''<sub>''k''</sub>}} are products of primes. But then {{math|1=''n'' = ''a b'' = ''p''<sub>1</sub> ''p''<sub>2</sub> ⋅⋅⋅ ''p''<sub>''j''</sub> ''q''<sub>1</sub> ''q''<sub>2</sub> ⋅⋅⋅ ''q''<sub>''k''</sub>}} is a product of primes.

===Uniqueness===
Suppose, to the contrary, there is an integer that has two distinct prime factorizations. Let {{math|''n''}} be the least such integer and write {{math|1=''n'' = ''p''<sub>1</sub> ''p''<sub>2</sub> ... ''p''<sub>''j''</sub> = ''q''<sub>1</sub> ''q''<sub>2</sub> ... ''q''<sub>''k''</sub>}}, where each {{math|''p''<sub>''i''</sub>}} and {{math|''q''<sub>''i''</sub>}} is prime. We see that {{math|''p''<sub>1</sub>}} divides {{math|''q''<sub>1</sub> ''q''<sub>2</sub> ... ''q''<sub>''k''</sub>}}, so {{math|''p''<sub>1</sub>}} divides some {{math|''q''<sub>''i''</sub>}} by [[Euclid's lemma]]. Without loss of generality, say {{math|''p''<sub>1</sub>}} divides {{math|''q''<sub>1</sub>}}. Since {{math|''p''<sub>1</sub>}} and {{math|''q''<sub>1</sub>}} are both prime, it follows that {{math|1=''p''<sub>1</sub> = ''q''<sub>1</sub>}}. Returning to our factorizations of {{math|''n''}}, we may cancel these two factors to conclude that {{math|1=''p''<sub>2</sub> ... ''p''<sub>''j''</sub> = ''q''<sub>2</sub> ... ''q''<sub>''k''</sub>}}. We now have two distinct prime factorizations of some integer strictly smaller than {{math|''n''}}, which contradicts the minimality of {{math|''n''}}.

===Uniqueness without Euclid's lemma===
The fundamental theorem of arithmetic can also be proved without using Euclid's lemma.<ref>{{citation |last1=Dawson |first1=John W. |title=Why Prove it Again? Alternative Proofs in Mathematical Practice. |date=2015 |publisher=Springer |isbn=9783319173689 |page=45}}</ref> The proof that follows is inspired by Euclid's original version of the [[Euclidean algorithm]]. 

Assume that <math>s</math> is the smallest positive integer which is the product of prime numbers in two different ways. Incidentally, this implies that <math>s</math>, if it exists, must be a [[composite number]] greater than <math>1</math>. Now, say 

:<math>
\begin{align}
s
&=p_1 p_2 \cdots p_m \\
&=q_1 q_2 \cdots q_n.
\end{align}
</math>

Every <math>p_i</math> must be distinct from every <math>q_j.</math> Otherwise, if say <math>p_i=q_j,</math> then there would exist some positive integer <math>t=s/p_i=s/q_j</math> that is smaller than {{mvar|s}} and has two distinct prime factorizations. One may also suppose that <math>p_1 < q_1,</math> by exchanging the two factorizations, if needed.

Setting <math>P=p_2\cdots p_m</math> and <math>Q=q_2\cdots q_n,</math> one has <math>s=p_1P=q_1Q.</math>
Also, since <math>p_1 < q_1,</math> one has <math>Q < P.</math>
It then follows that
:<math>s-p_1Q = (q_1-p_1)Q = p_1(P-Q) < s.</math>

As the positive integers less than {{mvar|s}} have been supposed to have a unique prime factorization, <math>p_1</math> must occur in the factorization of either <math>q_1-p_1</math> or {{mvar|Q}}. The latter case is impossible, as {{mvar|Q}}, being smaller than {{mvar|s}}, must have a unique prime factorization, and <math>p_1</math> differs from every <math>q_j.</math> The former case is also impossible, as, if <math>p_1</math> is a divisor of <math>q_1-p_1,</math> it must be also a divisor of <math>q_1,</math> which is impossible as <math>p_1</math> and <math>q_1</math> are distinct primes.

Therefore, there cannot exist a smallest integer with more than a single distinct prime factorization. Every positive integer must either be a prime number itself, which would factor uniquely, or a composite that also factors uniquely into primes, or in the case of the integer <math>1</math>, not factor into any prime.

==Generalizations==
{{more citations needed section|date=January 2024}}
The first generalization of the theorem is found in Gauss's second monograph (1832) on [[biquadratic reciprocity]]. This paper introduced what is now called the [[ring theory|ring]] of [[Gaussian integer]]s, the set of all [[complex number]]s ''a'' + ''bi'' where ''a'' and ''b'' are integers. It is now denoted by <math>\mathbb{Z}[i].</math> He showed that this ring has the four units ±1 and ±''i'', that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes ([[up to]] the order and multiplication by units).<ref>[[#CITEREFGauss1832|Gauss, BQ, §§ 31–34]]</ref>

Similarly, in 1844 while working on [[cubic reciprocity]], [[Gotthold Eisenstein|Eisenstein]] introduced the ring <math>\mathbb{Z}[\omega]</math>, where <math display="inline">\omega = \frac{-1 + \sqrt{-3}}{2},</math> &nbsp; <math>\omega^3 = 1</math> is a cube [[root of unity]]. This is the ring of [[Eisenstein integer]]s, and he proved it has the six units <math>\pm 1, \pm\omega, \pm\omega^2</math> and that it has unique factorization.

However, it was also discovered that unique factorization does not always hold. An example is given by <math>\mathbb{Z}[\sqrt{-5}]</math>. In this ring one has<ref>{{Harvtxt|Hardy|Wright|2008|loc=§ 14.6}}</ref>

:<math>
  6 =
  2 \cdot 3 =
  \left(1 + \sqrt{-5}\right)\left(1 - \sqrt{-5}\right).
</math>

Examples like this caused the notion of "prime" to be modified. In <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> it can be proven that if any of the factors above can be represented as a product, for example, 2 = ''ab'', then one of ''a'' or ''b'' must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; for example, 2 divides neither (1 + {{sqrt|&minus;5}}) nor (1 &minus; {{sqrt|&minus;5}}) even though it divides their product 6. In [[algebraic number theory]] 2 is called [[irreducible element|irreducible]] in <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> (only divisible by itself or a unit) but not [[prime element|prime]] in <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> (if it divides a product it must divide one of the factors). The mention of <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> is required because 2 is prime and irreducible in <math>\mathbb{Z}.</math> Using these definitions it can be proven that in any [[integral domain]] a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers <math>\mathbb{Z}</math> every irreducible is prime". This is also true in <math>\mathbb{Z}[i]</math> and <math>\mathbb{Z}[\omega],</math> but not in <math>\mathbb{Z}[\sqrt{-5}].</math>

The rings in which factorization into irreducibles is essentially unique are called [[unique factorization domain]]s. Important examples are [[polynomial ring]]s over the integers or over a [[field (mathematics)|field]], [[Euclidean domain]]s and [[principal ideal domain]]s.

In 1843 [[Ernst Kummer|Kummer]] introduced the concept of [[ideal number]], which was developed further by [[Richard Dedekind|Dedekind]] (1876) into the modern theory of [[Ideal (ring theory)|ideals]], special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called [[Dedekind domain]]s.

There is a version of [[ordinal arithmetic|unique factorization for ordinals]], though it requires some additional conditions to ensure uniqueness.

Any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Fundamental Theorem of Arithmetic is, in fact, a special case of the unique factorization theorem in commutative Möbius monoids.

==See also==
*[[Integer factorization]]
*[[List of theorems called fundamental]]
*[[Prime signature]], a characterization of how many primes divide a given number

==Notes==
{{reflist|30em}}

==References==
{{sfn whitelist |CITEREFHardyWright2008}}
The ''[[Disquisitiones Arithmeticae]]'' has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
* {{citation
  | last = Gauss  | first = Carl Friedrich
  | translator-last = Clarke | translator-first = Arthur A.
  | title = Disquisitiones Arithemeticae (Second, corrected edition)
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = New York
  | year = 1986
  | isbn = 978-0-387-96254-2
  | url = https://www.springer.com/mathematics/algebra/book/978-0-387-96254-2
}}
* {{citation
  | last = Gauss  | first = Carl Friedrich
  | translator-last = Maser | translator-first = H.
  | language = de
  | title = Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
  | publisher = Chelsea
  | location = New York
  | year = 1965
  | isbn = 0-8284-0191-8}}

The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § ''n''". Footnotes referencing the ''Disquisitiones Arithmeticae'' are of the form "Gauss, DA, Art. ''n''".
* {{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | title = Theoria residuorum biquadraticorum, Commentatio prima
  | publisher = Comment. Soc. regiae sci, Göttingen 6
  | location = Göttingen
  | year = 1828}}
* {{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | title = Theoria residuorum biquadraticorum, Commentatio secunda
  | publisher = Comment. Soc. regiae sci, Göttingen 7
  | location = Göttingen
  | year = 1832}}

These are in Gauss's ''Werke'', Vol II, pp.&nbsp;65–92 and 93–148; German translations are pp.&nbsp;511–533 and 534–586 of the German edition of the ''Disquisitiones''.
* {{citation
  | author1 = Euclid
  | author1-link = Euclid
  | others = Translated by [[Thomas Little Heath]]
  | title = The thirteen books of the Elements
  | edition = Second Edition Unabridged
  | volume = 2 (Books III-IX)
  | publisher = [[Dover]]
  | location = New York
  | year = 1956
  | isbn = 978-0-486-60089-5
  | url = https://archive.org/details/thirteenbooksofe00eucl
  | url-access = registration
  | ref = {{harvid|Euclid|Heath|1956}}
  }}
* {{Hardy and Wright|mode=cs2}}
* {{citation
 | first1 = Calvin T.
 | last1 = Long
 | year = 1972
 | title = Elementary Introduction to Number Theory
 | edition = 2nd | publisher = [[D. C. Heath and Company]]
 | location = Lexington
 | lccn = 77-171950
}}.
* {{citation
 | first1 = Anthony J.
 | last1 = Pettofrezzo
 | first2 = Donald R.
 | last2 = Byrkit
 | year = 1970
 | title = Elements of Number Theory
 | publisher = [[Prentice Hall]]
 | location = Englewood Cliffs
 | lccn = 77-81766
}}.
* {{citation
  | last1 = Riesel  | first1 = Hans
  | title = Prime Numbers and Computer Methods for Factorization (second edition)
  | publisher = Birkhäuser
  | location = Boston
  | year = 1994
  | isbn = 0-8176-3743-5}}
* {{citation |last= Weil |first= André |year= 2007 |orig-year= 1984 |title= [[Number Theory: An Approach through History from Hammurapi to Legendre]] |series= Modern Birkhäuser Classics |location= Boston, MA |publisher= Birkhäuser |isbn= 978-0-817-64565-6 }}

== External links ==
* [https://gowers.wordpress.com/2011/11/13/why-isnt-the-fundamental-theorem-of-arithmetic-obvious Why isn’t the fundamental theorem of arithmetic obvious?]
* [http://www.cut-the-knot.org/blue/gcd_fta.shtml GCD and the Fundamental Theorem of Arithmetic] at [[cut-the-knot]].
* [http://planetmath.org/fundamentaltheoremofarithmeticproofofthe PlanetMath: Proof of fundamental theorem of arithmetic]
* [http://fermatslasttheorem.blogspot.com/2005/06/unique-factorization.html Fermat's Last Theorem Blog: Unique Factorization], a blog that covers the history of [[Fermat's Last Theorem]] from [[Diophantus of Alexandria]] to the proof by [[Andrew Wiles]].
* [http://demonstrations.wolfram.com/FundamentalTheoremOfArithmetic/ "Fundamental Theorem of Arithmetic"] by Hector Zenil, [[Wolfram Demonstrations Project]], 2007.
* {{citation|last=Grime|first=James|title=1 and Prime Numbers|url=https://www.youtube.com/watch?v=IQofiPqhJ_s| archive-url=https://ghostarchive.org/varchive/youtube/20211211/IQofiPqhJ_s| archive-date=2021-12-11 | url-status=live|work=Numberphile|date=3 February 2012 |publisher=[[Brady Haran]]}}{{cbignore}}

{{Divisor classes}}

[[Category:Theorems about prime numbers]]
[[Category:Articles containing proofs]]
[[Category:Uniqueness theorems]]
[[Category:factorization]]

[[de:Primfaktorzerlegung#Fundamentalsatz der Arithmetik]]