From: jmchen@pub.jiangmen.gd.cn (Chen Shuwen) Subject: Great Progress on the Prouhet-Tarry-Escott problem! Date: 7 Sep 1999 14:44:03 -0400 Newsgroups: sci.math Keywords: Solution for k=11 Great Progress on the Prouhet-Tarry-Escott problem! Dear Professor Ideal solutions of the the Prouhet-Tarry-Escott problem has been solved for the k=11 case recently. That is [ 0, 11, 24, 65, 90, 129, 173, 212, 237, 278, 291, 302 ] = [ 3, 5, 30, 57, 104, 116, 186, 198, 245, 272, 297, 299 ] It means 0^k+11^k+24^k+65^k+90^k+129^k+173^k+212^k+237^k+278^k+291^k+302^k = 3^k+5^k+30^k+57^k+104^k+116^k+186^k+198^k+245^k+272^k+297^k+299^k (k=1,2,3,4,5,6,7,8,9,10,11) It is a great progress of the Prouhet-Tarry-Escott problem. (The first solution of the k=9 case was found in 1940 without using computer. and no new result of k>=10 is found in the past 60 years.) For more information, please visit the following site: http://member.netease.com/~chin/eslp/k1to11.htm http://member.netease.com/~chin/eslp/k246810.htm http://member.netease.com/~chin/eslp/eslp.htm http://member.netease.com/~chin/eslp/status.htm http://member.netease.com/~chin/eslp/TarryPrb.htm Best wishes, Chen Shuwen ============================================================================== From: gerry@mpce.mq.edu.au (Gerry Myerson) Subject: Re: Great Progress on the Prouhet-Tarry-Escott problem! Date: Wed, 08 Sep 1999 16:27:46 +1100 Newsgroups: sci.math In article , Allan Adler wrote: => There are general results on the Tarry-Escott problem, one of the earliest => being due to Prouhet. Prouhet's general result (early 1850's) has k => arbitrarily large. Yes, but neither Prouhet nor anyone else proves that there exist *ideal* solutions for all k. An ideal solution is one with only k + 1 terms on each side. Gerry Myerson (gerry@mpce.mq.edu.au) ============================================================================== From: jmchen@pub.jiangmen.gd.cn (Chen Shuwen) Subject: Ideal Solution of The Tarry-Escoot Problem Date: 8 Sep 1999 10:24:37 -0400 Newsgroups: sci.math arbitrarily large.Allan Adler write: >There are general results on the Tarry-Escott problem, one of >the earliest being due to Prouhet. Prouhet's general result (early >1850's) has k arbitrarily large.Allan Adler To the System a1^k+a2^k+...+am^k=b1^k+b2^k+...+bm^k (k=1,2,...n) Prouhet's general result is that there are m=2^n numbers on the left side, and m=2^k numbers on the righr side. When we discuss the Tarry-Escoot Problem, the most intereting is the ideal solution, where m=n+1. Please refer to my site: http://member.netease.com/~chin/eslp/TarryPrb.htm Best wishes. Chen Shuwen ============================================================================== From: jpr2718@aol.com (Jpr2718) Subject: Re: Tary-Escott problem. Date: 28 Apr 1999 00:28:46 GMT Newsgroups: sci.math blang@club-internet.fr (Bruno LANGLOIS) wrote: >Can we find 6 integers a,b,c,d,e,f such that : > >1) (a,b,c)<>(d,e,f) >2) a+b+c=d+e+f >3) a^2+b^2+c^2=d^2+e^2+f^2 >4) a^3+b^3+c^3=d^3+e^3+f^3 ? No. Newton's formulas can be used to give a simple proof. Or see Peter Borwein and Colin Ingalls, The Prouhet-Tarry-Escott Problem Revisited, Enseign.Math. (2).40.(1994),no.1-2,3-27 at: fttp://www.cecm.sfu.ca/~pborwein/PAPERS/P98.ps John ============================================================================== From: Dave Rusin Subject: Re: more specific problem Date: Wed, 27 Jan 1999 08:33:30 -0600 (CST) Newsgroups: [missing] To: stoimeno@informatik.hu-berlin.de >Restated, my problem was: is there for any k\in N a 0\ne m in Z >such that the polynomials {1+nx:n in Z} generate by products the >power series 1+mx^k+O(x^(k+1)). Now, what happens if we also allow >their inverse series 1/(1+nx)=1-nx+(nx)^2-... ? So you want two sets of integers {n_1i} and {n_2i} so that the products Prod(1+n_1i x) and Prod(1+n_2i x) have the first k terms the same -- is that right? I'm not really sure of the status of this, but it's essentially the Tarry-Escott "multigrades" problem. (That problem asks for the minimum size of such collections of integers, so I'm guessing that the _existence_ of these sets is easy to show?) Here are a couple of resources: index/11DXX.html points to 94/multigrades 96/multigrades There's also an "equal sums of like powers" page which is relevant here: http://www.nease.net/~chin/eslp/index.html dave