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单词 FundamentalTheoremOfArithmetic
释义

fundamental theorem of arithmetic


Each positive integer n has a unique as a productPlanetmathPlanetmath

n=i=0lpiai

of positive powers of its distinct positive prime divisorsPlanetmathPlanetmath pi. The prime divisor of n means a (rational) prime numberMathworldPlanetmath 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 thatpi<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.

Titlefundamental theorem of arithmetic
Canonical nameFundamentalTheoremOfArithmetic
Date of creation2013-03-22 11:46:03
Last modified on2013-03-22 11:46:03
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id21
AuthorCWoo (3771)
Entry typeTheorem
Classificationmsc 11A05
Classificationmsc 17B66
Classificationmsc 17B45
Related topicDivisibility
Related topicUFD
Related topicAnyNonzeroIntegerIsQuadraticResidue
Related topicNumberTheory
Definesprime divisor
Definesprime factor
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