View source for Fundamental theorem of arithmetic

{{short description|A positive integer factorizes uniquely into a product of primes}}
{{Distinguish|Fundamental theorem of algebra}}
[[File:Disqvisitiones-800.jpg|thumb|The unique factorization theorem was proved by [[Carl Friedrich Gauss|Gauss]] with his 1801 book ''[[Disquisitiones Arithmeticae]]''.<ref name="Gauss1801.loc=16">{{Harvtxt|Gauss|Clarke|1986|loc=Art. 16}}</ref> In this book, Gauss used the fundamental theorem for proving the [[law of quadratic reciprocity]].<ref>{{Harvtxt|Gauss|Clarke|1986|loc=Art. 131}}</ref>]]
In [[number theory]], the '''fundamental theorem of arithmetic''', also called the '''unique factorization theorem''' or the '''unique-prime-factorization theorem''', states that every [[integer]] greater than 1<ref>Using the [[empty product|empty product rule]] one need not exclude the number 1, and the theorem can be stated as: every positive integer has unique prime factorization.</ref> either is a [[prime number]] itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, [[up to]] (except for) 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,

:1200 = 2{{sup|4}} × 3{{sup|1}} × 5{{sup|2}} = 2 × 2 × 2 × 2 × 3 × 5 × 5 = 5 × 2 × 5 × 2 × 3 × 2 × 2 = ...

The theorem says two things for 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 (e.g., 12 = 2 × 6 = 3 × 4).

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, {{nowrap|1=2 = 2 × 1 = 2 × 1 × 1 = ...}}

==Euclid's original version==

Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' are essentially the statement and proof of the fundamental theorem.

{{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.

Article 16 of [[Carl Friedrich Gauss|Gauss]]' ''[[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 ''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 {{nowrap begin}}''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>k</sub>{{nowrap end}} are primes and the ''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 ''k'' = 0).

This representation is called the '''canonical representation'''<ref>{{harvtxt|Long|1972|p=45}}</ref> of ''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.

Note that factors ''p''<sup>0</sup> = 1 may be inserted without changing the value of ''n'' (e.g., 1000 = 2<sup>3</sup>×3<sup>0</sup>×5<sup>3</sup>).<br>In fact, any positive integer can be uniquely represented as an [[infinite product]] taken over all the positive prime numbers:
:<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 ''n''<sub>''i''</sub> are positive integers, and the rest 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 {{math|''p''}} [[Divisor|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''}}.

===Existence===
It must be shown that every integer greater than 1 is either prime or a product of primes.  First, 2 is prime.  Then, by [[strong induction]], assume this is true for all numbers greater than 1 and less than ''n''. If ''n'' is prime, there is nothing more to prove. Otherwise, there are integers ''a'' and ''b'',  where ''n'' = ''ab'', and {{nowrap|1 < ''a'' &le; ''b'' < ''n''}}.  By the induction hypothesis, {{nowrap|1=''a'' = ''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''j''</sub>}} and {{nowrap|1=''b'' = ''q''<sub>1</sub>''q''<sub>2</sub>...''q''<sub>''k''</sub>}} are products of primes. But then {{nowrap|1=''n'' = ''ab'' = ''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 ''n'' be the least such integer and write ''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 ''p''<sub>''i''</sub> and ''q''<sub>''i''</sub> is prime. (Note ''j'' and ''k'' are both at least 2.) We see ''p''<sub>1</sub> divides ''q''<sub>1</sub> ''q''<sub>2</sub> ... ''q''<sub>''k''</sub>, so ''p''<sub>1</sub> divides some ''q''<sub>''i''</sub> by Euclid's lemma. Without loss of generality, say ''p''<sub>1</sub> divides ''q''<sub>1</sub>. Since ''p''<sub>1</sub> and ''q''<sub>1</sub> are both prime, it follows that ''p''<sub>1</sub> = ''q''<sub>1</sub>. Returning to our factorizations of ''n'', we may cancel these two terms to conclude ''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 ''n'', which contradicts the minimality of ''n''.

===Elementary proof of uniqueness===
The fundamental theorem of arithmetic can also be proved without using Euclid's lemma, as follows:

Assume that ''s'' > 1 is the smallest positive integer which is the product of prime numbers in two different ways. If ''s'' were prime then it would factor uniquely as itself, so ''s'' is not prime and there must be at least two primes in each factorization of ''s'':

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

If any ''p''<sub>''i''</sub> = ''q''<sub>''j''</sub> then, by cancellation, ''s''/''p''<sub>''i''</sub> = ''s''/''q''<sub>''j''</sub> would be another positive integer, different from s, which is greater than 1 and also has two distinct factorizations. But ''s''/''p''<sub>''i''</sub> is smaller than ''s'', meaning ''s'' would not actually be the smallest such integer. Therefore every ''p''<sub>''i''</sub> must be distinct from every ''q''<sub>''j''</sub>.

Without loss of generality, take ''p''<sub>1</sub> < ''q''<sub>1</sub> (if this is not already the case, switch the ''p'' and ''q'' designations.) Consider

:<math>t = (q_1 - p_1)(q_2 \cdots q_n),</math>

and note that 1 < ''q''<sub>2</sub> ≤ ''t'' < ''s''. Therefore ''t'' must have a unique prime factorization. By rearrangement we see,

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

Here ''u'' = ((''p''<sub>2</sub> ... ''p''<sub>''m''</sub>) - (''q''<sub>2</sub> ... ''q''<sub>''n''</sub>)) is positive, for if it were negative or zero then so would be its product with ''p''<sub>''1''</sub>, but that product equals ''t'' which is positive. So ''u'' is either 1 or factors into primes. In either case, ''t'' = ''p''<sub>1</sub>''u'' yields a prime factorization of ''t'', which we know to be unique, so ''p''<sub>1</sub> appears in the prime factorization of ''t''.

If (''q''<sub>1</sub> - ''p''<sub>1</sub>) equaled 1 then the prime factorization of ''t'' would be all ''q'''s, which would preclude ''p''<sub>1</sub> from appearing. Thus  (''q''<sub>1</sub> - ''p''<sub>1</sub>) is not 1, but is positive, so it factors into primes: (''q''<sub>1</sub> - ''p''<sub>1</sub>) = (''r''<sub>1</sub> ... ''r''<sub>''h''</sub>). This yields a prime factorization of

:<math>t = (r_1 \cdots r_h)(q_2 \cdots q_n),</math>

which we know is unique. Now, ''p''<sub>1</sub> appears in the prime factorization of ''t'', and it is not equal to any ''q'', so it must be one of the ''r'''s. That means ''p''<sub>1</sub> is a factor of (''q''<sub>1</sub> - ''p''<sub>1</sub>), so there exists a positive integer ''k'' such that ''p''<sub>1</sub>''k'' = (''q''<sub>1</sub> - ''p''<sub>1</sub>), and therefore

:<math>p_1(k+1) = q_1.</math>

But that means ''q''<sub>1</sub> has a proper factorization, so it is not a prime number. This contradiction shows that ''s'' does not actually have two different prime factorizations. As a result, there is no smallest positive integer with multiple prime factorizations, hence all positive integers greater than 1 factor uniquely into primes.

==Generalizations==
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.<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>\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=
(1+\sqrt{-5})(1-\sqrt{-5}).
</math>

Examples like this caused the notion of "prime" to be modified. In <math>\mathbb{Z}[\sqrt{-5}]</math> it can be proven that if any of the factors above can be represented as a product, e.g., 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; e.g.,
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}[\sqrt{-5}]</math> (only divisible by itself or a unit) but not [[prime element|prime]] in <math>\mathbb{Z}[\sqrt{-5}]</math> (if it divides a product it must divide one of the factors). The mention of <math>\mathbb{Z}[\sqrt{-5}]</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.

==See also==
* [[Riemann zeta function#Euler product formula|Euler product formula]]
* [[Integer factorization]]
* [[Noetherian ring]]
* [[Prime signature]]

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

==References==

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.
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  | 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
}}
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  | publisher = Chelsea
  | location = New York
  | year = 1965
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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''".
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  | 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''.
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* {{Hardy and Wright}}
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* {{mathworld|urlname=AbnormalNumber|title=Abnormal number}}
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== 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.
* {{cite web|last=Grime|first=James|title=1 and Prime Numbers|url=http://www.numberphile.com/videos/1notprime.html|work=Numberphile|publisher=[[Brady Haran]]}}

{{Fundamental theorems}}
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[[Category:Fundamental theorems|Arithmetic]]

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