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fundamental theorem of arithmetic
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(Theorem)
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Each positive integer  has a unique decomposition as a product
of positive powers of its distinct positive prime divisors  . The prime divisor of  means a (rational) prime number dividing  . A synonymous name is prime factor.
The decomposition is unique up to the order of the prime divisors and for  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' 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.
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Cross-references: Euclidean, egyptian fraction, properties, congruences, arithmetic, Gauss, empty product, prime number, rational, product, integer, positive
There are 82 references to this entry.
This is version 15 of fundamental theorem of arithmetic, born on 2001-10-15, modified 2008-01-02.
Object id is 221, canonical name is FundamentalTheoremOfArithmetic.
Accessed 12698 times total.
Classification:
| AMS MSC: | 11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors) |
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Pending Errata and Addenda
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