   #copyright

Fundamental theorem of arithmetic

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   In number theory, the fundamental theorem of arithmetic (or unique
   factorization theorem) states that every natural number greater than 1
   can be written as a unique product of prime numbers. For instance,

          6936 = 2^3 \times 3 \times 17^2 , \,\!

          1200 = 2^4 \times 3 \times 5^2 . \,\!

   There are no other possible factorizations of 6936 or 1200 into prime
   numbers. The above representation collapses repeated prime factors into
   powers for easier identification. Because multiplication is
   commutative, the order of factors is irrelevant and usually written
   from smallest to largest.

   Many authors take the natural numbers to begin with 0, which has no
   prime factorization. Thus Theorem 1 of Hardy & Wright (1979) takes the
   form, “Every positive integer, except 1, is a product of primes”, and
   Theorem 2 (their "Fundamental") asserts uniqueness. The number 1 is not
   itself prime, but since it is the product of no numbers, it is often
   convenient to include it in the theorem by the empty product rule.
   (See, for example, Calculating the GCD.)

Applications

   The fundamental theorem of arithmetic establishes the importance of
   prime numbers. Prime numbers are the basic building blocks of any
   positive integer, in the sense that each positive integer can be
   constructed from the product of primes with one unique construction.
   Finding the prime factorization of an integer allows derivation of all
   its divisors, both prime and non-prime.

   For example, the above factorization of 6936 shows that any positive
   divisor of 6936 must have the form 2^a × 3^b × 17^c, where a takes one
   of the 4 values in {0, 1, 2, 3}, where b takes one of the 2 values in
   {0, 1}, and where c takes one of the 3 values in {0, 1, 2}. Multiplying
   the numbers of independent options together produces a total of 4 × 2 ×
   3 = 24 positive divisors.

   Once the prime factorizations of two numbers are known, their greatest
   common divisor and least common multiple can be found quickly. For
   instance, from the above it is shown that the greatest common divisor
   of 6936 and 1200 is 2^3 × 3 = 24. However if the prime factorizations
   are not known, the use of the Euclidean algorithm generally requires
   much less calculation than factoring the two numbers.

   The fundamental theorem ensures that additive and multiplicative
   arithmetic functions are completely determined by their values on the
   powers of prime numbers.

Proof

   The theorem was essentially first proved by Euclid, but the first full
   and correct proof is found in the Disquisitiones Arithmeticae by Carl
   Friedrich Gauss.

   Although at first sight the theorem seems 'obvious', it does not hold
   in more general number systems, including many rings of algebraic
   integers. This was first pointed out by Ernst Kummer in 1843, in his
   work on Fermat's last theorem. The recognition of this failure is one
   of the earliest developments in algebraic number theory.

Euclid's proof

   The proof consists of two steps. In the first step every number is
   shown to be a product of zero or more primes. In the second, the proof
   shows that any two representations may be unified into a single
   representation.

Non-prime composite numbers

   Suppose there were a positive integer which cannot be written as a
   product of primes. Then there must be a smallest such number: let it be
   n. This number n cannot be 1, because of the convention above. It
   cannot be a prime number either, since any prime number is a product of
   a single prime, itself. So it must be a composite number. Thus

          n = ab

   where both a and b are positive integers smaller than n. Since n is the
   smallest number which cannot be written as a product of primes, both a
   and b can be written as products of primes. But then

          n = ab

   can be written as a product of primes as well, a contradiction. This is
   a minimal counterexample argument.

Proof by infinite descent

   A proof of the uniqueness of the prime factorization of a given integer
   uses infinite descent: Assume that a certain integer can be written as
   (at least) two different products of prime numbers, then there must
   exist a smallest integer s with such a property. Denote these two
   factorizations of s as p[1] ... p[m] and q[1] ... q[n], such that

   s = p[1]p[2] ... p[m] = q[1]q[2] ... q[n].

   No p[i] (with 1 ≤ i ≤ m) can be equal to any q[j] (with 1 ≤ j ≤ n), as
   there would otherwise be a smaller integer factorizable in two ways (by
   removing prime factors common in both products) violating the above
   assumption. Now it can be assumed without loss of generality that p[1]
   is a prime factor smaller than any q[j] (with 1 ≤ j ≤ n). Take q[1].
   Then there exist integers d and r such that

          q[1]/p[1] = d + r/p[1]

   and 0 < r < p[1] < q[1] (r can't be 0, as that would make q[1] a
   multiple of p[1] and not prime). Multiplying both sides by s / q[1],
   the result is

          p[2] ... p[m] = (d + r/p[1]) q[2] ... q[n] = dq[2] ... q[n] +
          rq[2] ... q[n]/p[1].

   The second term in the last expression must be equal to an integer,
   which can be called k, i.e.

          k = rq[2] ... q[n]/p[1].

   This gives

          p[1]k = rq[2] ... q[n].

   The value of both sides of this equation is obviously smaller than s,
   but is still large enough to be factorizable. Since r is smaller than
   p[1], the two prime factorizations we get on each side after both k and
   r are written out as their product of primes must be different. This is
   in contradiction with s being the smallest integer factorizable in more
   than one way. Thus the original assumption must be false.

Proof using abstract algebra

   Let n be an integer. Z[n] is a finite group and therefore has a
   composition series. By definition, the factors in a composition series
   are simple. Hence the factors in a composition series of Z[n] are of
   the form Z[p] for some prime number p. Since the order of Z[n] is the
   product of the orders of the factors of the composition series, this
   gives a factorization of n into prime numbers. But the Jordan-Hölder
   theorem says that our composition series is unique, and hence the prime
   factorization of n must be unique.

   Retrieved from "
   http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
   of authors and sources) and is available under the GNU Free
   Documentation License. See also our Disclaimer.
