Fundamental theorem of arithmetic

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 is either prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.[1][2][3] For example,

$1200 = 2^4 \times 3^1 \times 5^2 = 3 \times 2\times 2\times 2\times 2 \times 5 \times 5 = 5 \times 2\times 3\times 2\times 5 \times 2 \times 2 =\cdots\text { etc.} \!$

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).

History

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

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

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

This representation is called the canonical representation[4] of n, or the standard form[5][6] 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

This representation is convenient for expressions like these for the product, gcd, and lcm:

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

While expressions like these are of great theoretical importance their practical use is limited by our ability to factor numbers.

Arithmetical functions

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). The article has proofs of the lemma.

Existence

By induction: assume it is true for all numbers less than n. If n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < ab < n. By the induction hypothesis, a = p1p2...pn and b = q1q2...qm are products of primes. But then n = ab = p1p2...pnq1q2...qm is the product of primes. In the base case, 2 is a trivial 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 all m of the pi, showing that mn. If there were any qj left over we would have

\begin{align} \frac{s}{p_1 p_2 \cdots p_m} &=1 \\ &=q_k \cdots q_n, \end{align}

which is impossible, since the product of numbers greater than 1 cannot equal 1. Therefore m = n and every qj is a pi.

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.[7]

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[8]

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

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 (only divisible by itself or a unit) but not prime (if it divides a product it must divide one of the factors). 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 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.

Notes

1. ^ Long (1972, p. 44)
2. ^ Pettofrezzo & Byrkit (1970, p. 53)
3. ^ Hardy & Wright, Thm 2
4. ^ Long (1972, p. 45)
5. ^ Pettofrezzo & Byrkit (1970, p. 55)
6. ^ Hardy & Wright § 1.2
7. ^ Gauss, BQ, §§ 31–34
8. ^ Hardy & Wright, § 14.6

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 0387962549
• Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber 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
• Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (fifth ed.), USA: Oxford University Press, ISBN 978-0-19-853171-5
• 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