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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[note 1] either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors.[3][4][5] For example,

1200 = 24 × 31 × 52 = 3 × 2 × 2 × 2 × 2 × 5 × 5 = 5 × 2 × 3 × 2 × 5 × 2 × 2 = etc.

The theorem is stating two things: first, that 1200 can be represented as a product of primes, and second, no matter how this is done, there will always be 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 numbers may not be unique (e.g. 12 = 2 × 6 = 3 × 4).

This theorem is one of the main reasons for which 1 is not considered as a prime number: if 1 would be prime, the factorization would not be unique, as, for example, 2 = 2×1 = 2×1×1 = ...

History

Book VII, propositions 30 and 32 of Euclid's Elements is essentially the statement and proof of the fundamental theorem.

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, Elements Book VII, Proposition 30

Proposition 30 is referred to as Euclid's lemma. And it is the key in the proof of the fundamental theorem of arithmetic.

Any composite number is measured by some prime number.

— Euclid, Elements Book VII, Proposition 31

Proposition 31 is derived from proposition 30.

Any number either is prime or is measured by some prime number.

— Euclid, Elements Book VII, Proposition 32

Proposition 32 is derived from proposition 31.

Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.[1]
Applications
Canonical representation of a positive integer

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

$$n = p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_k^{\alpha_k} = \prod_{i=1}^{k}p_i^{\alpha_i}$$

where p1 < p2 < ... < pk are primes and the αi are positive integers.

This representation is called the canonical representation[6] of n, or the standard form[7][8] of n.

For example 999 = 33×37, 1000 = 23×53, 1001 = 7×11×13

Note that factors p0 = 1 may be inserted without changing the value of n (e.g. 1000 = 23×30×53).
In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers,

$$n=2^{n_2}3^{n_3}5^{n_5}7^{n_7}\cdots=\prod p_i^{n_{p_i}},$$

where a finite number of the ni are positive integers, and the rest are zero. Allowing negative exponents provides a canonical form for positive rational numbers.
Arithmetic operations

The canonical representation, when it is known, is convenient for easily computing products, gcd, and lcm:

$$a\cdot b =2^{a_2+b_2}\,3^{a_3+b_3}\,5^{a_5+b_5}\,7^{a_7+b_7}\cdots =\prod$$ p_i^{a_{p_i}+b_{p_i}},

$$\gcd(a,b) =2^{\min(a_2,b_2)}\,3^{\min(a_3,b_3)}\,5^{\min(a_5,b_5)}\,7^{\min(a_7,b_7)}\cdots =\prod p_i^{\min(a_{p_i},b_{p_i})},$$

$$\operatorname{lcm}(a,b) =2^{\max(a_2,b_2)}\,3^{\max(a_3,b_3)}\,5^{\max(a_5,b_5)}\,7^{\max(a_7,b_7)}\cdots =\prod p_i^{\max(a_{p_i},b_{p_i})}.$$

However, as Integer factorization of large integers is much harder than computing their product, gcd or lcm, these formulas have, in practice, a limited usage.

Arithmetical functions
Main article: Arithmetic function

Many arithmetical functions are defined using the canonical representation. In particular, the values of additive and 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 p divides the product of two natural numbers a and b, then p divides a or p divides b (or both).

Existence

We need to show that every integer greater than 1 is a product of primes. By induction: assume it is true for all numbers between 1 and n. If n is prime, there is nothing more to prove (a prime is a trivial product of primes, a "product" with only one factor). Otherwise, there are integers a and b, where n = ab and 1 < ab < n. By the induction hypothesis, a = p1p2...pj and b = q1q2...qk are products of primes. But then n = ab = p1p2...pjq1q2...qk is a product of primes.

Uniqueness

Assume that s > 1 is the product of prime numbers in two different ways:

\begin{align} s &=p_1 p_2 \cdots p_m \\ &=q_1 q_2 \cdots q_n. \end{align}

We must show m = n and that the qj are a rearrangement of the pi.

By Euclid's lemma, p1 must divide one of the qj; relabeling the qj if necessary, say that p1 divides q1. But q1 is prime, so its only divisors are itself and 1. Therefore, p1 = q1, so that

\begin{align} \frac{s}{p_1} &=p_2 \cdots p_m \\ &=q_2 \cdots q_n. \end{align}

Reasoning the same way, p2 must equal one of the remaining qj. Relabeling again if necessary, say p2 = q2. Then

\begin{align} \frac{s}{p_1 p_2} &=p_3 \cdots p_m \\ &=q_3 \cdots q_n. \end{align}

This can be done for each of the m pi's, showing that m ≤ n and every pi is a qj. Applying the same argument with the p's and q's reversed shows n ≤ m (hence m = n) and every qj is a pi.
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 there must be at least two primes in each factorization of s:

\begin{align} s &=p_1 p_2 \cdots p_m \\ &=q_1 q_2 \cdots q_n. \end{align}

If any pi = qj then, by cancellation, s/pi = s/qj would be a positive integer greater than 1 with two distinct factorizations. But s/pi is smaller than s, meaning s would not actually be the smallest such integer. Therefore every pi must be distinct from every qj.

Without loss of generality, take p1 < q1 (if this is not already the case, switch the p and q designations.) Consider

$$t = (q_1 - p_1)(q_2 \cdots q_n),$$

and note that 1 < q2 ≤ t < s. Therefore t must have a unique prime factorization. By rearrangement we see,

\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}

Here u = ((p2 ... pm) - (q2 ... qn)) is positive, for if it were negative or zero then so would be its product with p1, but that product equals t which is positive. So u is either 1 or factors into primes. In either case, t = p1u yields a prime factorization of t, which we know to be unique, so p1 appears in the prime factorization of t.

If (q1 - p1) equaled 1 then the prime factorization of t would be all q's, which would preclude p1 from appearing. Thus (q1 - p1) is not 1, but is positive, so it factors into primes: (q1 - p1) = (r1 ... rh). This yields a prime factorization of

$$t = (r_1 \cdots r_h)(q_2 \cdots q_n),$$

which we know is unique. Now, p1 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 p1 is a factor of (q1 - p1), so there exists a positive integer k such that p1k = (q1 - p1), and therefore

$$p_1(k+1) = q_1.$$

But that means q1 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 of Gaussian integers, the set of all complex numbers a + bi where a and b are integers. It is now denoted by $$\mathbb{Z}[i]$$. 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.[9]

Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring $$\mathbb{Z}[\omega]$$, where $$\omega=\frac{-1+\sqrt{-3}}{2}, \omega^3=1$$ is a cube root of unity. This is the ring of Eisenstein integers, and he proved it has the six units $$\pm 1, \pm\omega, \pm\omega^2$$ and that it has unique factorization.

However, it was also discovered that unique factorization does not always hold. An example is given by $$\mathbb{Z}[\sqrt{-5}]$$. In this ring one has[10]

$$6= 2\times 3= (1+\sqrt{-5})\times(1-\sqrt{-5}).$$

Examples like this caused the notion of "prime" to be modified. In $$\mathbb{Z}[\sqrt{-5}]$$ 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 + √−5) nor (1 − √−5) even though it divides their product 6. In algebraic number theory 2 is called irreducible in \mathbb{Z}[\sqrt{-5}] (only divisible by itself or a unit) but not prime in in $$\mathbb{Z}[\sqrt{-5}]$$ (if it divides a product it must divide one of the factors). The mention of $$\mathbb{Z}[\sqrt{-5}]$$ is required because 2 is prime and irreducible in $$\mathbb{Z}$$. Similarly, 5 is prime and irreducible in \mathbb{Z} and not prime nor irreducible in $$\mathbb{Z}[\sqrt{-5}]$$. Using these definitions it can be proven that in any ring a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers $$\mathbb{Z} every irreducible is prime". This is also true in \( \mathbb{Z}[i]$$ and $$\mathbb{Z}[\omega]$$, but not in $$\mathbb{Z}[\sqrt{-5}]$$.

The rings where every irreducible is prime are called unique factorization domains. As the name indicates, the fundamental theorem of arithmetic is true in them. Important examples are polynomial rings over the integers or over a field, Euclidean domains and principal ideal domains.

In 1843 Kummer introduced the concept of ideal number, which was developed further by Dedekind (1876) into the modern theory of ideals, special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains.

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

Euler product formula
Integer factorization
Noetherian ring
Prime signature

Notes

Gauss & Clarke (1986, Art. 16)
Gauss & Clarke (1986, Art. 131)
Long (1972, p. 44)
Pettofrezzo & Byrkit (1970, p. 53)
Hardy & Wright (2008, Thm 2)
Long (1972, p. 45)
Pettofrezzo & Byrkit (1970, p. 55)
Hardy & Wright (2008, § 1.2)
Gauss, BQ, §§ 31–34

Hardy & Wright (2008, § 14.6)

Using the empty product rule one need not exclude the number 1, and the theorem can be stated as: every positive integer has unique prime factorization.

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.

Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 978-0-387-96254-2
Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, 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".

Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6
Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7

These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148; German translations are pp. 511–533 and 534–586 of the German edition of the Disquisitiones.

Baker, Alan (1984), A Concise Introduction to the Theory of Numbers, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-28654-1
Euclid (1956), The thirteen books of the Elements, 2 (Books III-IX), Translated by Thomas Little Heath (Second Edition Unabridged ed.), New York: Dover, ISBN 978-0-486-60089-5
Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001
A. Kornilowicz; P. Rudnicki (2004), "Fundamental theorem of arithmetic", Formalized Mathematics 12 (2): 179–185
Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950.
Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766.
Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 0-8176-3743-5
Weisstein, Eric W., "Abnormal number", MathWorld.
Weisstein, Eric W., "Fundamental Theorem of Arithmetic", MathWorld.