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Unformatted text preview: 5 7 > 0, we can express log 5 7 as a fraction p q with p,q ∈ N . log 5 7 = p q ⇒ 5 p q = 7 ⇒ 5 p = 7 q . Since 5 divides the lefthandside, 5 must also divide the righthandside. (Notice that we are using the fact that p ∈ N to conclude 5  5 p and this is not true for p ≤ 0.) Therefore, we have that 5  7 q ⇒ 5  7 ··· 7  {z } q times ⇒ 5  7 , (By Chap . 4 . 3 , Lemma 3) which is a contradiction as 7 is a prime. Therefore, our initial hypothesis must have been wrong, and hence log 5 7 is irrational. 2...
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 Fall '12
 Gupta
 Math, Prime number, Greatest common divisor, Continued fraction

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