From: jmchen@pub.jiangmen.gd.cn (Chen Shuwen)
Newsgroups: sci.math.numberthy
Subject: Welcome to Chen Shuwen's Equal Sums of Like Powers Site
Date: 4 Aug 97 14:30:00 GMT

Welcome to Chen Shuwen's Equal Sums of Like Powers Site
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Dear Professor,

I am Chen Shuwen, a young man of P.R.China. I graduated from Zhongshan
(Sun Yat-sen) University six years ago.

I have great interested on the following number theory problem (Equal
Sums of Like Powers, The Prouhet-Tarry-Escott Problem) and had studied
it for about 11 years.

        a_1^k+a_2^k+...+a_m^k=b_1^k+b_2^k+...+b_m^k
                where (k=k_1,k_2,...,k_n).

For example:

  1^1+19^1+20^1+51^1+57^1+80^1+82^1 = 2^1+12^1+31^1+40^1+69^1+71^1+85^1
  1^2+19^2+20^2+51^2+57^2+80^2+82^2 = 2^2+12^2+31^2+40^2+69^2+71^2+85^2
  1^3+19^3+20^3+51^3+57^3+80^3+82^3 = 2^3+12^3+31^3+40^3+69^3+71^3+85^3
  1^4+19^4+20^4+51^4+57^4+80^4+82^4 = 2^4+12^4+31^4+40^4+69^4+71^4+85^4
  1^5+19^5+20^5+51^5+57^5+80^5+82^5 = 2^5+12^5+31^5+40^5+69^5+71^5+85^5
  1^6+19^6+20^6+51^6+57^6+80^6+82^6 = 2^6+12^6+31^6+40^6+69^6+71^6+85^6

  975^2+224368^2+300495^2+366448^2 = 37648^2+202575^2+337168^2+344655^2
  975^3+224368^3+300495^3+366448^3 = 37648^3+202575^3+337168^3+344655^3
  975^4+224368^4+300495^4+366448^4 = 37648^4+202575^4+337168^4+344655^4

It is known so far that:

<1> When m=n, no non-negative integer solution is found to this system.

<2> When m=n, integer solutions are found to the following 10 types:
(k=1,2,4) (k=1,2,6) (k=1,2,3,5) (k=1,2,4,6) (k=1,2,3,5,7)
(k=1,2,3,4,6) (k=1,2,4,6,8) (k=1,2,3,4,5,7) (k=1,2,3,4,5,6,8)
(k=1,2,3,4,5,6,7,9)
 *Three types among the above ten are solved by me first.

<3> When m=n+1,non-negative integer solutions are found to the
following 32 types:
(k=1) (k=2) (k=3) (k=4) (k=1,2) (k=1,3) (k=1,4) (k=1,5) (k=2,3) (k=2,4)
(k=2,6) (k=1,2,3) (k=1,2,4) (k=1,2,5) (k=1,2,6) (k=1,3,4) (k=1,3,5)
(k=2,3,4) (k=2,4,6) (k=1,2,3,4) (k=1,2,3,5) (k=1,2,4,6) (k=1,3,5,7)
(k=12,4,6,8) (k=1,2,3,4,5) (k=1,2,3,4,6) (k=1,2,4,6,8) (k=1,2,3,4,5,6)
(k=1,2,3,4,5,7) (k=1,2,3,4,5,6,7) (k=1,2,3,4,5,6,7,8)
(k=1,2,3,4,5,6,7,8,9)
*10 types among the above 32 types are solved by me first.

Now my homepage on equal sums of like powers is finished, it contains
almost all the important numerical results on this subject. Please visit
and introduce to your friend who may have interested on it.

There are about 50 pages on my site. Here are some main URL:

1)Chen Shuwen's Equal Sums of Like Powers Site (Homapage)
http://www.jiangmen.gd.cn/person/chen/chenhome.htm

2)Discussion and comments (E-mails sent to Chen discussing the topic "
Equal sums of like powers" or commenting on Chen's site. In this page
you also can see that my site has been linked to several Number Thoery
Web in the world)
http://www.jiangmen.gd.cn/person/chen/discuss.htm

3)Unsolved problems and conjectures (On equal sums of like powers)
http://www.jiangmen.gd.cn/person/chen/unsolve.htm

If you like my pages or have interested on this subject, please mail me.

Best regards


Chen Shuwen
==================================================
Jiang Xing Electronics Limited,
138#, Shengli Road,
Jiangmen City, Guangdong Province, 529000
People's Republic of China
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Tel: 0750-3793136 (Home), 01392886443 (Mobile)
E-Mail: jmchen@pub.jiangmen.gd.cn
Homepage: (Equal Sums of Like Powers Site)
http://www.jiangmen.gd.cn/person/chen/chenhome.htm
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