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