Difference between revisions of "Fundamental Theorem of Arithmetic"
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| − | The '''Fundamental Theorem of Arithmetic''' states that every [[positive integer]] <math>n</math> can be written as a product <math>n = p_1 \cdot p_2 \cdot \cdots \cdot p_k</math> where the <math>p_i</math> are all [[prime number]]s; moreover, this [[expression]] for <math>n</math> (called its [[prime factorization]]) is unique, up to rearrangement of the factors. | + | The '''Fundamental Theorem of Arithmetic''' states that every [[positive integer]] <math>n</math> can be written as a product <math>n = p_1 \cdot p_2 \cdot \cdots \cdot p_k</math> where the <math>\displaystyle p_i</math> are all [[prime number]]s; moreover, this [[expression]] for <math>n</math> (called its [[prime factorization]]) is unique, up to rearrangement of the factors. |
Note that the property of uniqueness is not, in general, true for other sorts of factorizations. For example, most integers have many factorizations into 2 parts: <math>30 = 2 \cdot 15 = 3 \cdot 10 = 5 \cdot 6</math>. Thus, the Fundamental Theorem of Arithmetic tells us in some sense that "factorizations into prime numbers is deeper than factorization into two parts." | Note that the property of uniqueness is not, in general, true for other sorts of factorizations. For example, most integers have many factorizations into 2 parts: <math>30 = 2 \cdot 15 = 3 \cdot 10 = 5 \cdot 6</math>. Thus, the Fundamental Theorem of Arithmetic tells us in some sense that "factorizations into prime numbers is deeper than factorization into two parts." | ||
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Revision as of 18:36, 16 January 2007
The Fundamental Theorem of Arithmetic states that every positive integer 
can be written as a product 
where the 
are all prime numbers; moreover, this expression for (called its prime factorization) is unique, up to rearrangement of the factors.
Note that the property of uniqueness is not, in general, true for other sorts of factorizations. For example, most integers have many factorizations into 2 parts: 
. Thus, the Fundamental Theorem of Arithmetic tells us in some sense that "factorizations into prime numbers is deeper than factorization into two parts."
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