esrever
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Report Thread starter 3 years ago
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I was solving a problem. Stumbled upon a totally irrelevant problem. Found an awesome proof for it. The proof is very similar to another famous proof (won't say which one). Try it!

Prove that there don't exist any non-integer rational numbers a,b such that a-b, a+b, ab are all integers.

EDIT: apparently an easier proof exists too
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Zacken
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Assume that a,b are non-integer rationals with a-b, a+b, ab all integer. Then p=2a and q=2b are integers and have to be odd. So pq is odd. But we know that pq/4 is an integer so pq is divisible by four. Contradiction.
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esrever
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(Original post by Zacken)
Assume that a,b are non-integer rationals with a-b, a+b, ab all integer. Then p=2a and q=2b are integers and have to be odd. So pq is odd. But we know that pq/4 is an integer so pq is divisible by four. Contradiction.
Yea. I found a much more indirect proof which was similar to proving \sqrt{2} is irrational. Basically assume a,b are rationals in simplest form and then arrive at contradiction.
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