# 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 $p_{i}$. 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 $p_{i} whenever $i.

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