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Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds:

For all $a,b$ with $a^2+b^2

What is the least such $n$?

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    Just curiosity: where did this problem come up?2012-03-06
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    @J.D. In my mind, out of curiosity aswell.2012-03-06
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    I'm thinking of the *very* special case of $n = 3$ and $p_1, p_2, p_n$ forming an equilateral triangle of unit side length. Let $(a,b)$ be the center of the circle passing through $p_1, p_2, p_3$. Then for $1\le i\le 3$, the distance from $(a,b)$ to $p_i$ is $\dfrac{\sqrt{3}}{3}$. But again very special.2012-03-06
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    Oops. Just noticed "*for all* $a,b$...". Nevermind my comment above.2012-03-06
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    Do you really mean $\forall a,b\in B(0,\sqrt{e}),\ |(a,b)-p_i|$ is irrational? Because that is certainly false. The set of points rational distance from a given point $p\in\mathbb{R}^2$ is dense in $\mathbb{R}^2$.2012-03-07
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    @Neal No, I meant what I wrote2012-03-07
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    This seems like a difficult problem, and perhaps there are no such collections of points in general. You might also want to consider the question "What is the largest such $n$?"2012-03-07
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    @you Infinity? put points at y=0,x=1,2,3,4,5,6,....2012-03-07
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    Something like this is discussed in Problem D19 of Guy, Unsolved Problems In Number Theory. Almering proved that the points at rational distances from the vertices of any triangle with rational edges are dense in the plane; I believe this shows $n\ge4$. It may be in MR0147447 (26 #4963) Almering, J. H. J. Rational quadrilaterals. Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math. 25 1963 192–199.2012-03-07
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    @Holowitz Why was that not what you wrote? Are you assuming that $a$ and $b$ are integers?2012-03-07
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    @Neal For all a,b, there is atleast one i (which may depend/wary with a and b), such that the distance is irrational2012-03-07
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    OH. Sorry, being stupid today.2012-03-07
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    @Gerry: Another paper cited in that problem, which proves the slightly stronger result that the points at rational distances from the vertices of a triangle are dense in the plane if at least one length is rational and the squares of all lengths are rational, is available online: [T. G. Berry, Points at a rational distance from the vertices of a triangle, *Acta Arith.*, **62** (1992) 391-398](http://matwbn.icm.edu.pl/ksiazki/aa/aa62/aa6246.pdf).2012-03-07
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    The Unsolved Problems book is also online [here](http://www.futuretg.com/FTHumanEvolutionCourse/FTFreeLearningKits/01-MA-Mathematics%20and%20Preparation%20for%20University/011-MA11-UN03-10-Number%20Theory%20and%20Cryptography/Additional%20Resources/Richard%20K%20Guy%20-%20Unsolved%20Problems%20In%20Number%20Theory%202ed.pdf).2012-03-07
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    I asked this here http://mathoverflow.net/questions/90454/least-cardinality-of-a-set-of-points-in-the-plane2012-03-07

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(To kick it from the unanswered queue)

This question is asked and answered on MO (as mentioned by the OP): Least cardinality of a set of points in the plane


Tony Huynh's solution:

As Boris Bukh points out, three points suffice, but I'd like to point out that your question is related to this MO question.

Here is a summary of the information in the previous question. For the second part of your question, the author (me) conjectures that for any finite set $S$ with all rational distances, no such point $P$ exists. As I noted in the comments, this is true when $|S|=3$, proven by Almering.

It is not known if there is a point with all rational distances to the unit square. However, it is known that there are no points at rational distance from all vertices of a regular $n$-gon, except perhaps when $n=4,6,8,12,24$.

Some more tangential remarks are that it is not known if there is a dense set of points in the plane with all distances rational, although it is conjectured that there is none.

Even more tangential, it is not known if every planar graph can be straight-line embedded in the plane with all edges having rational length, although it is conjectured to be true.