Unformatted text preview: n D 1 ; 1 cannot be written as the product of any nonempty collection of prime numbers. So consider a natural number n ³ 2 and assume inductively that the assertion of unique factorization holds for all natural numbers less than n . Consider two factorizations of n as before, and assume without loss of generality that q 1 ± p 1 . Since q 1 divides n D p 1 p 2 :::p s , it follows from Proposition 1.6.17 that q 1 divides, and hence is equal to, one of the p i . Since also q 1 ± p 1 ± p k for all k , it follows that p 1 D q 1 . Now dividing both sides by q 1 , we get n=q 1 D q 2 :::q r ; n=q 1 D p 2 :::p s :...
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 Fall '08
 EVERAGE
 Algebra, Integers, Prime number, nonzero integers

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