fundamental theorem of arithmetic

Each positive integer n has a unique as a productPlanetmathPlanetmath


of positive powers of its distinct positive prime divisorsPlanetmathPlanetmath pi. The prime divisor of n means a (rational) prime numberMathworldPlanetmath dividing ( 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 pi<pj whenever i<j.

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 congruencesMathworldPlanetmathPlanetmathPlanetmathPlanetmath. 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 fractionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath arithmetic. Fibonacci, for example, wrote in Egyptian fraction arithmetic, used three notations to detail EuclideanMathworldPlanetmath 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