From rusin@math.niu.edu Mon Mar 29 11:20:52 1999 Received: from vesuvius.math.niu.edu (vesuvius.math.niu.edu [131.156.3.93]) by clinch.math.niu.edu (8.9.1a/8.9.1) with ESMTP id LAA00562; Mon, 29 Mar 1999 11:20:51 -0600 (CST) From: Dave Rusin Received: (from rusin@localhost) by vesuvius.math.niu.edu (8.8.8/8.8.5) id LAA19224; Mon, 29 Mar 1999 11:20:51 -0600 (CST) Date: Mon, 29 Mar 1999 11:20:51 -0600 (CST) Message-Id: <199903291720.LAA19224@vesuvius.math.niu.edu> To: powersum@chez.com Subject: Re: http://www.chez.com/powersum/ Cc: rusin@math.niu.edu Status: RO There appear to be some errors in the data at this site, if I understand it correctly. You show a table which begins with the following entry: power (k) = 2 number of left terms (m) = 1 There is in this entry the number "2"; I assume you mean that "it is known that it is possible to find examples of a square which can be expressed as a sum of 2 nonzero squares". Of course this statement is true. This number is printed with a dark blue background. According to your table, this means " number of right terms (n), best lower bound conjectured" Perhaps you have the color wrong? But this isn't consistent with what I see for k=5, m=1. The "4" there should mean "it is known that it is possible to find examples of a fifth power which can be expressed as a sum of 4 nonzero fifth powers, BUT it is not known whether or not it is possible to find examples of a fifth power which can be expressed as a sum of 3 nonzero fifth powers." I am unable to understand why the term "6" for k=10,m=6 is given in red; if you have two equal sums of six 10th powers, that is useful, and yet it is not proven to be minimal. So calling this "number of right terms (n) that doesn't need to be explored" seems wrong. Finally, on page http://www.chez.com/powersum/records.htm, I don't understand your references. For example, Elkies's example (4,1,3) is about 1986, not 1995. I can't understand what Allan MacLeod,1998 is supposed to tell us, since Elkies actually showed there are infinitely many (4,1,3) solutions. Why list just one more? You should also state at some point that you are interested in sums of powers of _positive_ integers. Certainly sums like 1^5=0^5+0^5+1^5 are not interesting. Also you seem to distinguish the problems a^5+b^5=c^5+d^5 and a^5+b^5+c^5=d^5; they _are_ different problems if you insist a,b,c,d be positive, but they are the same problem if you simply ask that they be nonzero integers. dave PS You may wish to refer the reader to index/11PXX.html or to some of the references on that page. From graph@worldnet.fr Mon Mar 29 16:04:49 1999 Received: from Bespin.worldnet.net (bespin.worldnet.net [195.3.3.4]) by clinch.math.niu.edu (8.9.1a/8.9.1) with ESMTP id QAA02346 for ; Mon, 29 Mar 1999 16:04:47 -0600 (CST) Received: from default (p14-102.province.worldnet.fr [195.3.14.102]) by Bespin.worldnet.net (8.8.8/8.8.8) with SMTP id AAA19109 for ; Tue, 30 Mar 1999 00:02:33 +0200 (CEST) Message-ID: <004201be7a30$0ca03f40$660e03c3@default> From: "Jean-Charles Meyrignac" To: "Dave Rusin" Subject: Re: http://www.chez.com/powersum/ Date: Mon, 29 Mar 1999 23:57:31 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 4.72.3110.5 X-MimeOLE: Produced By Microsoft MimeOLE V4.72.3110.3 Content-Transfer-Encoding: 7bit Status: RO Hello ! > >There appear to be some errors in the data at this site, if I understand it >correctly. > Ok, let's see this... > >You show a table which begins with the following entry: >power (k) = 2 >number of left terms (m) = 1 >There is in this entry the number "2"; I assume you mean that "it is known >that it is possible to find examples of a square which can be expressed >as a sum of 2 nonzero squares". Of course this statement is true. >This number is printed with a dark blue background. According to your table, >this means >" > number of right terms (n), best lower bound conjectured" >Perhaps you have the color wrong? > >But this isn't consistent with what I see for k=5, m=1. The "4" there >should mean "it is known that it is possible to find examples of a >fifth power which can be expressed as a sum of 4 nonzero fifth powers, >BUT it is not known whether or not it is possible to find examples of a >fifth power which can be expressed as a sum of 3 nonzero fifth powers." > You are right, but I didn't figure how to make the things clearer (my english is not very good, and I'm not a mathematician, and I tried to make my index page 'attractive'). I have put a new color on the powers below 5, with the legend: number of right terms (n), best lower bound proved Can you suggest a better formulation ? > >I am unable to understand why the term "6" for k=10,m=6 is given in red; >if you have two equal sums of six 10th powers, that is useful, and yet >it is not proven to be minimal. So calling this >"number of right terms (n) that doesn't need to be explored" >seems wrong. > I had this kind of question when I put the (10,6,6) cell in red. During 2 years, I thought that (10,7,7) was optimal, then Randy Ekl found (10,6,6) and Bob Scher told me about the k=m+n conjecture. Now, I strongly believe that we should have k=m+n, so we should have two equal sums of five 10th powers. Of course, this is a belief and it might be wrong. But, instead of conjecturing, we are computing... I hope that the current project will shortly find (10,5,6) -I'm currently computing (10,6,6) with a new algorithm-. > >Finally, on page http://www.chez.com/powersum/records.htm, I don't >understand your references. For example, Elkies's example (4,1,3) is >about 1986, not 1995. > I corrected the reference, thanks ! > > I can't understand what Allan MacLeod,1998 is >supposed to tell us, since Elkies actually showed there are infinitely >many (4,1,3) solutions. Why list just one more? > Elkies found a general formula, but the 3 referenced equations are the lowest known. > >You should also state at some point that you are interested in sums of >powers of _positive_ integers. Certainly sums like 1^5=0^5+0^5+1^5 are >not interesting. Also you seem to distinguish the problems >a^5+b^5=c^5+d^5 and a^5+b^5+c^5=d^5; they _are_ different problems if >you insist a,b,c,d be positive, but they are the same problem if you >simply ask that they be nonzero integers. > You are right, I modified the first terms of my index (I put: positive non-zero integers). > >PS You may wish to refer the reader to >index/11PXX.html >or to some of the references on that page. > I had already put your link on my links page www.chez.com/powersum/links.htm ) I just simply modified the reference of your link. Don't hesitate to warn me about errors ! Thanks. Jean-Charles From graph@worldnet.fr Mon Mar 29 16:09:25 1999 Received: from Bespin.worldnet.net (bespin.worldnet.net [195.3.3.4]) by clinch.math.niu.edu (8.9.1a/8.9.1) with ESMTP id QAA02369 for ; Mon, 29 Mar 1999 16:09:23 -0600 (CST) Received: from default (p14-102.province.worldnet.fr [195.3.14.102]) by Bespin.worldnet.net (8.8.8/8.8.8) with SMTP id AAA20279 for ; Tue, 30 Mar 1999 00:07:14 +0200 (CEST) Message-ID: <000601be7a30$b3f43ee0$660e03c3@default> From: "Jean-Charles Meyrignac" To: "Dave Rusin" Subject: Re: http://www.chez.com/powersum/ Date: Tue, 30 Mar 1999 00:08:27 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 4.72.3110.5 X-MimeOLE: Produced By Microsoft MimeOLE V4.72.3110.3 Content-Transfer-Encoding: 7bit Status: RO > >PS You may wish to refer the reader to >index/11PXX.html >or to some of the references on that page. > PS: Can you put a link for our project on your page ? Thanks !