User:Virginia-American/Sandbox/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 (existence) that every integer greater than 1 is either prime itself or is the product of prime numbers, and (uniqueness) that, although the order of the primes in the second case is arbitrary, the primes themselves are not. 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 content of the theorem is that in any representation of 1200 as a product of primes, there will always be four 2s, one 3, and two 5s.

Classification of integers
To clarify the role of the integer 1, and to prepare for more general settings than the integers, it is useful to classify the integers using terminology from abstract algebra, specifically from algebraic number theory and ring theory:


 * zero 0
 * positive integers 1, 2, 3, ...
 * negative integers ..., &minus;3, &minus;2, &minus;1
 * units &minus;1 and 1
 * prime numbers ..., &minus;3, &minus;2, 2, 3, 5, ...
 * composite numbers ..., &minus;6, &minus;4, 4, 6, 8, 9, ...

Using it we have:

"A nonzero integer is either positive or negative." "A negative integer is the unit &minus;1 times a positive integer." "A positive integer is either the unit 1, a positive prime, or the product of positive primes. Up to the order of the factors, this product is uniquely determined by the integer." To avoid constantly repeating the special cases, the definition of "product" can be slightly expanded to include as "products" the two cases in wheich there is no actual multiplication: the empty product with no factors and the "product" with only one factor. Under this convention the theorem reads:

"Every positive integer is the product of positive primes, and, except for the order of the factors, in one way only. The integer 1 is the empty product of no primes."

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.

Canonical representation of a positive integer
Every positive integer n 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 of n, or the standard form of n. E.g., 12 = 22·3, 1296 = 24·34, and 220 = 22·5·11.

Note that factors p0 = 1 may be inserted without changing the value of n. In fact, any integer can be uniquely represented as an infinite product taken over all the prime numbers,

n=2^{n_2}3^{n_3}5^{n_5}7^{n_7}\cdots=\prod p_i^{n_{p_i}}. $$ where all but a finite number of the ni are zero.

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 perhaps both). The article has proofs of the lemma.

Existence
By inspection, the small natural numbers 1, 2, 3, 4, ... are the product of primes. This is the basis for a proof 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 < a ≤ b < n. By the induction hypothesis, a = p1p2...pn and b = q1q2...qm are products of primes. But then n = ab = p1p2...pnq1q2...qm is also.

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 m ≤ n. 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} $$

a contradiction, 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 on biquadratic reciprocity. This paper introduced what is now denoted $$\mathbb{Z}[i]$$, where $$i^2=-1.$$ This is the ring of Gaussian integers, and is the set of all complex numbers a + bi where a and b are integers. Gauss 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.

Similarly, Eisenstein introduced the ring $$\mathbb{Z}[\omega]$$, where  $$\omega=\frac{-1+\sqrt{-3}}{2}, $$   $$\omega^3=1.$$ This is the ring of Eisenstein integers, and Eisenstein proved its units are the six numbers $$\pm 1, \pm\omega, \pm\omega^2$$ 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

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

and all of factors can be proven prme in that ring.

In algebraic number theory, and more generally in ring theory, a unique factorization domain is defined as an algebraic structure in which the fundamental theorem of arithmetic holds. For example, any Euclidean domain or principal ideal domain can be proven to be a unique factorization domain.

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