[**BASE Cinque** - *Appunti di
Matematica ricreativa*]

a distanze reciproche razionali (o intere)

**Traduzione
e commento del teorema di Erdos-Anning**

**INTEGRAL
DISTANCES**

**NORMAN
H. ANNING AND PAUL ERDÖS**

In the present note we are going to prove the following result:

*For any n we can find n points in the plane not all on a line such that
their distances are all integral, but it is impossible to find infinitely many
points with integral distances (not all on a line).*

PROOF. Consider the circle of diameter 1,
*x*^{2}+*y*^{2}=1/4.

Let *p*_{1}, *p*_{2}, ... be the sequence of
primes of the form 4*k*+1.

It is well known that

*p _{i}*

is solvable.

Consider the point (on the circle
*x*^{2}+*y*^{2}=1/4) whose distance from (-1/2,
0) is *b _{i}*/

Denote this point by (*x _{i}*,

Consider the sequence of points (-1/2, 0), (1/2, 0),
(*x _{i}*,

We shall show that any two distances are rational.

Suppose this has been shown for all *i*<*j*.

We then prove that the distance from (*x _{j}*,

Consider the 4 concyclic points (-1/2, 0), (1/2, 0),
(*x _{j}*,

This completes the proof.

Thus of course by enlarging the radius of the circle we can obtain n points with integral distances.

It is very likely that these points are dense in the circle
*x*^{2}+*y*^{2}=1/4, but this we can not
prove.

*Data creazione: * marzo 2013

*Ultimo aggiornamento: * marzo 2013

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