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There is an infinite number of points in a plane. Devise an arrangement of these points in the plane in such a way that the distance between any two points in the plane is an integer.

I realise that this is proved in the Erdős-Anning theorem; however, I didn’t quite understand the proof. I know the answer is a straight line, but is there a simpler way to prove it without using sets and all that? (I’m an A-level student, so these concepts are a little hard for me to grasp.)

gen-ℤ ready to perish
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Saad Shaikh
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  • Generally speaking, Erdős tended to put a lot of effort into finding the most elegant or straightforward proofs for things he was interested in. In this case the proof requires little more than high school geometry and careful counting, although the specific writeup you have read may not make this clear. Given this, I believe it would be more productive to work on understanding the existing proof rather than avoid it and look for a completely different approach. It would help if you linked to the proof you read and told us what was hard to follow. – Erick Wong Oct 02 '17 at 17:52
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    I’ve suggested an edit that slightly alters a bit of your punctuation, but most importantly features the question in a block quote. If it gets approved and you feel it’s too picky (the perfect word slipped my mind), then feel free to roll it back $\ddot\smile$ – gen-ℤ ready to perish Oct 02 '17 at 19:30

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