Unformatted text preview: a b where a and b have no common factors. Then we have 3 = a 2 b 2 or 3 b 2 = a 2 . Thus 3  a 2 and by the above lemma 3  a . Let a = 3 k for some k ∈ Z . Thus 3 b 2 = (3 k ) 2 = 9 k 2 and dividing by 3 we get b 2 = 3 k 2 . Note that 3  b 2 and by the above lemma 3  b . Thus b = 3 l for some l ∈ Z . In conclusion a = 3 k and b = 3 l and thus a and b have the common factor 3. This contradicts what we assumed at the beginning and completes the proof. 1...
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 Spring '09
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 Math, Logic, Mathematical logic, 3k

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