From: gwoegi@fmatbds01.tu-graz.ac.REMOVE.at (Gerhard J. Woeginger)
Subject: Re: integer distances
Date: Wed, 3 Nov 1999 23:22:06
Newsgroups: sci.math.research
Keywords: points in the plane which are integral distances apart
In article <7vps4n$fgk$1@news.si.fct.unl.pt> "manuel" writes:
>From: "manuel"
>Subject: integer distances
>Date: Wed, 3 Nov 1999 17:46:01 -0000
> I was reading "Excursions in number theory"( ogilvy) from dover where
>the autor( they are in fact 2) p.69 he describes a curious method to obtain:
> an arbitary number of points in the plane at integral distances each from
>each (and not all lying in a line).
>Is it possible to have an infinite number of points satisfying those
>conditions?
No, this is not possible. That's a classical result by Anning and Erdoes.
("Integral distances", Bull. Am. Math. Soc. 51, 598-600, 1945).
- Gerhard
___________________________________________________________
Gerhard J. Woeginger (gwoegi@opt.math.tu-graz.ac.at)
==============================================================================
From: Robin Chapman
Subject: Re: integer distances
Date: Thu, 04 Nov 1999 08:51:14 GMT
Newsgroups: sci.math.research
In article <7vps4n$fgk$1@news.si.fct.unl.pt>,
"manuel" wrote:
> I was reading "Excursions in number theory"( ogilvy) from dover where
> the autor( they are in fact 2) p.69 he describes a curious method to obtain:
>
> an arbitary number of points in the plane at integral distances each from
> each (and not all lying in a line).
>
> Is it possible to have an infinite number of points satisfying those
> conditions?
No. There's a simple argument due, I believe, to Kaplansky.
Let A, B and C be three of the points which are not collinear.
Then for any other of the points P we have -|AB| <= |PA| - |PB| <= |AB|
and -|BC| <= |PB| - |PC| <= |BC|. Thus |PA| - |PB| and |PB| - |PC|
are each restricted to a finite range of values. But for each
permissible value P must lie on the intersection of two hyperbolas.
These meet in at most 4 points and so the number of possible |P| is
finite; indeed at most 4(1 + 2|AB|)(1 + 2|BC|).
--
Robin Chapman
http://www.maths.ex.ac.uk/~rjc/rjc.html
"`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
Greg Egan, _Teranesia_
Sent via Deja.com http://www.deja.com/
Before you buy.
==============================================================================
[Note that an accompanying paper by Erdos was reproduced _in its entirety_
in Mathematical Reviews! --djr]
7,164a 48.0X
Anning, Norman H.; Erdös, Paul
Integral distances.
Bull. Amer. Math. Soc. 51, (1945). 598--600.
The authors show that for any $n$ there exist noncollinear points
$P\sb 1,\cdots,P\sb n$ in the plane such that all distances $P\sb
iP\sb j$ are integers; but there does not exist an infinite set of
non-collinear points with this property. [Cf. the following review.]
Reviewed by I. Kaplansky
_________________________________________________________________
7,164b 48.0X
Erdös, Paul
Integral distances.
Bull. Amer. Math. Soc. 51, (1945). 996.
The paper reads as follows.
"In a note under the same title [see the preceding review] it was
shown that there does not exist in the plane an infinite set of
noncollinear points with all mutual distances integral.
"It is possible to give a shorter proof of the following
generalization: if $A,B,C$ are three points not in line and
$k=[\max\,(AB,BC)]$, then there are at most $4(k+1)\sp 2$ points $P$
such that $PA-PB$ and $PB-PC$ are integral. For $\vert PA-PB\vert $ is
at most $AB$ and therefore assumes one of the values $0,1,\cdots,k$,
that is, $P$ lies on one of $k+1$ hyperbolas. Similarly $P$ lies on
one of the $k+1$ hyperbolas determined by $B$ and $C$. These
(distinct) hyperbolas intersect in at most $4(k+1)\sp 2$ points. An
analogous theorem clearly holds for higher dimensions."
Reviewed by I. Kaplansky
© Copyright American Mathematical Society 2000