# Points in the plane at integer distances

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 $a^2+b^2<e$, there exists an integer $i$, with $1\leq i\leq n$, such that the distance from $(a,b)$ to $p_i$ is irrational.

What is the least such $n$?

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Just curiosity: where did this problem come up? – user2468 Mar 6 '12 at 16:44
@J.D. In my mind, out of curiosity aswell. – user1708 Mar 6 '12 at 16:52
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. – user2468 Mar 6 '12 at 17:15
Oops. Just noticed "for all $a,b$...". Nevermind my comment above. – user2468 Mar 6 '12 at 17:18
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. – Gerry Myerson Mar 7 '12 at 6:09