COT3100NumTheory03 - Fundamental Theorem of Arithmetic Even...

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Fundamental Theorem of Arithmetic Even though this is one of the most important results in all of Number Theory, it is rarely included in most high school syllabi (in the US) formally. Interestingly enough, almost everyone has an intuitive notion of this result and it is almost always informally covered in middle school mathematics classes in the United States. The Fundamental Theorem of Arithmetic simply states that each positive integer has an unique prime factorization. What this means is that it is impossible to come up with two distinct multisets of prime integers that both multiply to a given positive integer. To prove this, we must show two things: 1) Each positive integer can be prime factorized. 2) Each prime factorization is unique. To see the first fact, let m>1 be the smallest positive integer which does NOT have a prime factorization. Since m is not a prime number, we can write m as a product of two factors, m 1 and m 2 . But, since both of these are smaller than m, they DO have prime factorizations. Thus, m can be expressed as the product of these two factorizations, which creates a prime factorization contradicting the assumption that m does not have one.
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Before I continue with the second part of this proof, I want to introduce pi notation, which is very similar to sigma notation: ) ( * ) 1 ( * ... * ) 3 ( * ) 2 ( * ) 1 ( ) ( 1 n f n f f f f i f n i - = = The only difference between pi notation and sigma notation is that each designated term is multiplied instead of added. Also, I want to prove the two following lemmas: 1) If p is prime and a and b are positive integers and p | ab, then either p | a or p | b. 2) If p is prime and a i for 1 i n, are positive integers, and if p | a 1 a 2 ... a n then p | a i for some 1 i n. Proof of #1: If p | a, we are done. Now consider the situation where p does NOT divide evenly into a. In this case, we must have that gcd(p, a) = 1. (This gcd can not be p, and p has no other possible factors, so the gcd must be 1.) Thus, there exist
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COT3100NumTheory03 - Fundamental Theorem of Arithmetic Even...

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