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"Prove that if n is a perfect square, $\,n+2\,$ is NOT a perfect square." I'm having trouble picking a method to prove this. Would contraposition be a good option (or even work for that matter)? If not, how about contradiction?

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    The first question is not "What method shall I use?" but "What's happening here?". So fool around with numbers. Anyway, contraposition is probably hard on the back.2012-09-23
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    @AndréNicolas I disagree re. contraposition. To get from a square to the next smallest square, you have to subtract an odd number ...2012-09-23
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    @Neal: Ultimately we will use, at least implicitly, some logical apparatus. A little experimentation will show that the next square is always too much bigger, or some parity condition is violated. Once it is clear that this seems to be the case, one can worry about proving it. But the insight about what might be going wrong comes first. Does the Math Department have a new building? It has been forever since I was at IU.2012-09-23
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    @AndréNicolas I'm sorry, I should have been clearer. I wholeheartedly agree that insight should come first. I just meant that some insights would suggest logical contraposition. Anyway, the Math Dept. is still in Rawles, the grad students are still infesting Swain East, but the Atwater house got a makeover last year. Who did you work with at IU?2012-09-23
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    I was an Assistant Professor there. And we were in Swain. Some fond memories.2012-09-23

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