Fundamental theorem of arithmetic

Short description: Integers have unique prime factorizations

In Disquisitiones Arithmeticae (1801) Gauss proved the unique factorization theorem and used it to prove the law of quadratic reciprocity.

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 numbers, up to the order of the factors. For example,

[math]\displaystyle{ 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 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 numbers may not be unique (for example, [math]\displaystyle{ 12 = 2 \cdot 6 = 3 \cdot 4 }[/math]).

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

The theorem generalizes to other algebraic structures that are called unique factorization domains and include principal ideal domains, Euclidean domains, and polynomial rings over a field. However, the theorem does not hold for algebraic integers. 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 Fermat's statement and 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 Elements.

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

(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.

Any composite number is measured by some prime number.

– Euclid, 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 infinite descent.

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, and proves that the decomposition is possible.

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, 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. 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 and stated for the first time the fundamental theorem of arithmetic.

Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.

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

[math]\displaystyle{ n = p_1^{n_1}p_2^{n_2} \cdots p_k^{n_k} = \prod_{i=1}^{k} p_i^{n_i}, }[/math]

where p₁ < p₂ < ... < p_k are primes and the n_i 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 of n, or the standard form of n. For example,

999 = 3³×37,

1000 = 2³×5³,

1001 = 7×11×13.

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

[math]\displaystyle{ 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_i are positive integers, and the others are zero.

Allowing negative exponents provides a canonical form for positive rational numbers.

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]\displaystyle{ \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

Many arithmetic 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 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 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 = a b, and 1 < a ≤ b < n. By the induction hypothesis, a = p₁ p₂ ⋅⋅⋅ p_j and b = q₁ q₂ ⋅⋅⋅ q_k are products of primes. But then n = a b = p₁ p₂ ⋅⋅⋅ p_j q₁ q₂ ⋅⋅⋅ q_k 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 = p₁ p₂ ... p_j = q₁ q₂ ... q_k, where each p_i and q_i is prime. We see that p₁ divides q₁ q₂ ... q_k, so p₁ divides some q_i by Euclid's lemma. Without loss of generality, say p₁ divides q₁. Since p₁ and q₁ are both prime, it follows that p₁ = q₁. Returning to our factorizations of n, we may cancel these two factors to conclude that p₂ ... p_j = q₂ ... q_k. We now have two distinct prime factorizations of some integer strictly smaller than n, which contradicts the minimality of n.

Uniqueness without Euclid's lemma

The fundamental theorem of arithmetic can also be proved without using Euclid's lemma. The proof that follows is inspired by Euclid's original version of the Euclidean algorithm.

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

[math]\displaystyle{ \begin{align} s &=p_1 p_2 \cdots p_m \\ &=q_1 q_2 \cdots q_n. \end{align} }[/math]

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

Setting [math]\displaystyle{ P=p_2\cdots p_m }[/math] and [math]\displaystyle{ Q=q_2\cdots q_n, }[/math] one has [math]\displaystyle{ s=p_1P=q_1Q. }[/math] Also, since [math]\displaystyle{ p_1 \lt q_1, }[/math] one has [math]\displaystyle{ Q \lt P. }[/math] It then follows that

[math]\displaystyle{ s-p_1Q = (q_1-p_1)Q = p_1(P-Q) \lt s. }[/math]

As the positive integers less than s have been supposed to have a unique prime factorization, [math]\displaystyle{ p_1 }[/math] must occur in the factorization of either [math]\displaystyle{ q_1-p_1 }[/math] or Q. The latter case is impossible, as Q, being smaller than s, must have a unique prime factorization, and [math]\displaystyle{ p_1 }[/math] differs from every [math]\displaystyle{ q_j. }[/math] The former case is also impossible, as, if [math]\displaystyle{ p_1 }[/math] is a divisor of [math]\displaystyle{ q_1-p_1, }[/math] it must be also a divisor of [math]\displaystyle{ q_1, }[/math] which is impossible as [math]\displaystyle{ p_1 }[/math] and [math]\displaystyle{ 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]\displaystyle{ 1 }[/math], not factor into any prime.

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 [math]\displaystyle{ \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).

Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring [math]\displaystyle{ \mathbb{Z}[\omega] }[/math], where [math]\displaystyle{ \omega = \frac{-1 + \sqrt{-3}}{2}, }[/math]   [math]\displaystyle{ \omega^3 = 1 }[/math] is a cube root of unity. This is the ring of Eisenstein integers, and he proved it has the six units [math]\displaystyle{ \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]\displaystyle{ \mathbb{Z}[\sqrt{-5}] }[/math]. In this ring one has

[math]\displaystyle{ 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]\displaystyle{ \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 + √−5) nor (1 − √−5) even though it divides their product 6. In algebraic number theory 2 is called irreducible in [math]\displaystyle{ \mathbb{Z}\left[\sqrt{-5}\right] }[/math] (only divisible by itself or a unit) but not prime in [math]\displaystyle{ \mathbb{Z}\left[\sqrt{-5}\right] }[/math] (if it divides a product it must divide one of the factors). The mention of [math]\displaystyle{ \mathbb{Z}\left[\sqrt{-5}\right] }[/math] is required because 2 is prime and irreducible in [math]\displaystyle{ \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]\displaystyle{ \mathbb{Z} }[/math] every irreducible is prime". This is also true in [math]\displaystyle{ \mathbb{Z}[i] }[/math] and [math]\displaystyle{ \mathbb{Z}[\omega], }[/math] but not in [math]\displaystyle{ \mathbb{Z}[\sqrt{-5}]. }[/math]

The rings in which factorization into irreducibles is essentially unique are called unique factorization domains. 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.

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 – Decomposition of a number into a product
• Prime signature – Multiset of prime exponents in a 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 (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 978-0-387-96254-2, https://www.springer.com/mathematics/algebra/book/978-0-387-96254-2 
• Gauss, Carl Friedrich (1965) (in de), 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.

• 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, https://archive.org/details/thirteenbooksofe00eucl 
• 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. 
• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company .
• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall .
• Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 0-8176-3743-5 
• Weil, André (2007), Number Theory: An Approach through History from Hammurapi to Legendre, Modern Birkhäuser Classics, Boston, MA: Birkhäuser, ISBN 978-0-817-64565-6 

External links

• Why isn’t the fundamental theorem of arithmetic obvious?
• GCD and the Fundamental Theorem of Arithmetic at cut-the-knot.
• PlanetMath: Proof of fundamental theorem of arithmetic
• 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.
• "Fundamental Theorem of Arithmetic" by Hector Zenil, Wolfram Demonstrations Project, 2007.

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