fundamental theorem of arithmetic

Each positive integer $n$ has a unique as a product

$$n=\prod _{i=0}^{l}p_i{}^{a_i}$$

of positive powers of its distinct positive prime divisors $p_i$. The prime divisor of $n$ means a (rational) prime number dividing (http://planetmath.org/Divisibility) $n$. A synonymous name is prime factor.

The of the prime divisors and for  $n=1$  is an empty product.

For some results it is useful to assume that whenever .

The FTA was the last of the fundamental theorems proven by C.F. Gauss. Gauss wrote his proof in “Discussions on Arithmetic” (Disquisitiones Arithmeticae) in 1801 formalizing congruences. Euclid and Greeks used prime properties of the FTA without rigorously proving its existence. It appears that the fundamentals of the FTA were used centuries before, and after, the Greeks within Egyptian fraction arithmetic. Fibonacci, for example, wrote in Egyptian fraction arithmetic, used three notations to detail Euclidean and medieval factoring methods.

Title	fundamental theorem of arithmetic
Canonical name	FundamentalTheoremOfArithmetic
Date of creation	2013-03-22 11:46:03
Last modified on	2013-03-22 11:46:03
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	21
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 11A05
Classification	msc 17B66
Classification	msc 17B45
Related topic	Divisibility
Related topic	UFD
Related topic	AnyNonzeroIntegerIsQuadraticResidue
Related topic	NumberTheory
Defines	prime divisor
Defines	prime factor