This project is dedicated to all those who are fascinated by powers and integers.

In the following, k, m, n and every term a_i, b_j always denote positive integers.
For given k and m, this page summarizes all the known minimal solutions for n of the equation:

Lander, Parkin and Selfridge conjectured in 1966 that:

Given the power k and the left number of terms m, we are trying to lower the known right number of terms n.
You can find more informations on the detailed page .
If you want to participate, go to the download page .
You can check the project status here.
Here are the newsletter 1, newsletter 2

27th January 2000: New home page: http://euler.free.fr !
26th January 2000: Using the newest beta 9, Nuutti Kuosa found (10,2,12) in 2 hours !
112¹⁰+99¹⁰=109¹⁰+103¹⁰+83¹⁰+79¹⁰+72¹⁰+59¹⁰+59¹⁰+52¹⁰+20¹⁰+15¹⁰+5¹⁰+5¹⁰
25th January 2000: Euler2000 beta 9 is now available, and since ALL Resta's programs are here, client/server is now the priority !
4th January 2000: Euler2000 beta 7 is currently in test and a lot of new solutions have already been found !
Check this page for more results
Also, here are two new links, the first one is my software company Quantic Dream and the second one is from a participant: mathonline.cjb.net (very nice name)
1th January 2000: Happy new year to everybody ! And yes, this is our first birthday ! The project started one year ago.
Thanks to your work, we already found some great results, and 2000 will see more results. Thanks !
30th November: Larry Hays found a new solution to (6,2,5):
20111⁶+15059⁶=18879⁶+15680⁶+14854⁶+13812⁶+2499⁶
27th November: Corrected some bugs in Euler2000 beta 2.
NEW ! (6,2,5) search has started ! A new solution to (6,2,5) has already been found by Larry Hays and Tommy Nolan !
6623⁶+323⁶ = 6615⁶+2912⁶+642⁶+434⁶+363⁶
22th November: Corrected some bugs in Euler2000 beta 1.
21th November: The updates are less frequent, since I'm working onto Euler2000.
You can download the beta if you want to check the differences.
(6,2,5) Resta's implementation -slightly improved- now added, new AVL algorithm very soon.
7th November: Nuutti Kuosa found a new solution to (7,4,4):
1105⁷+666⁷+431⁷+314⁷=1098⁷+752⁷+536⁷+130⁷=2072756588947872048896
6th November 1999: Two highly recommended links:
http://pobox.com/~djb/sortedsums.html (D. J. Bernstein)
http://www.nease.net/~chin/eslp/ (Chen Shuwen)
2nd November 1999: Kevin O'Hare found (13,1,36):
990¹³=989¹³+709¹³+423¹³+305¹³+225¹³+157¹³+119¹³+98¹³+73¹³+52¹³+45¹³+37¹³*2+36¹³*2+31¹³+30¹³+28¹³+27¹³+25¹³+24¹³*4+22¹³*2+20¹³+16¹³+15¹³*2+9¹³+7¹³*2+3¹³*3
1st November 1999: Check the beautiful article of Roger E. Frye how he discovered (4,1,3) in 1988. Very instructive !
 Finding 95800^4 + 217519^4 + 414560^4 = 422481^4 (on the site, courtesy of R. Frye)

Older results can be found here

The following remarkable results were found by this project:

(6,2,5) 1117⁶+770⁶=1092⁶+861⁶+602⁶+212⁶+84⁶ (Edward Brisse and Giovanni Resta, 04/20/1999)
(6,2,5) 6623⁶+323⁶=6615⁶+2912⁶+642⁶+434⁶+363⁶ (Larry Hays and Tommy Nolan, 11/27/1999)
(6,2,5) 20111⁶+15059⁶=18879⁶+15680⁶+14854⁶+13812⁶+2499⁶ (Larry Hays, 11/30/1999)
(7,1,7) 568⁷=525⁷+439⁷+430⁷+413⁷+266⁷+258⁷+127⁷ (Mark Dodrill, 03/20/1999)
(7,4,4) 343⁷+281⁷+46⁷+35⁷=354⁷+112⁷+52⁷+19⁷ (Michael Lau, 05/12/1999)
(7,4,4) 554⁷+412⁷+365⁷+146⁷=536⁷+439⁷+437⁷+65⁷ (Nuutti Kuosa, 09/05/1999)
(9,2,10) 137⁹+69⁹=121⁹+116⁹*2+115⁹+89⁹+52⁹+28⁹+26⁹+14⁹+9⁹ (Luigi Morelli, 03/12/1999)
(10,1,14) 120¹⁰=115¹⁰+104¹⁰+91¹⁰+80¹⁰*2+74¹⁰+63¹⁰+55¹⁰+40¹⁰+35¹⁰+33¹⁰+13¹⁰+10¹⁰+5¹⁰ (Torbjörn Alm, 10/14/1999)
(10,2,12) 112¹⁰+99¹⁰=109¹⁰+103¹⁰+83¹⁰+79¹⁰+72¹⁰+59¹⁰+59¹⁰+52¹⁰+20¹⁰+15¹⁰+5¹⁰+5¹⁰ (Nuutti Kuosa, 01/26/2000)
(10,4,12) 53¹⁰+44¹⁰*2+22¹⁰=51¹⁰+49¹⁰+43¹⁰+39¹⁰+29¹⁰+28¹⁰+17¹⁰*2+16¹⁰+13¹⁰+7¹⁰+4¹⁰ (Eric Bainville)
(10,6,6) 151¹⁰+140¹⁰+127¹⁰+86¹⁰+61¹⁰+22¹⁰=148¹⁰+146¹⁰+121¹⁰+94¹⁰+47¹⁰+35¹⁰ (Nuutti Kuosa, 09/03/1999)
(11,1,16) 78¹¹=71¹¹+69¹¹+66¹¹+64¹¹+63¹¹+52¹¹+47¹¹+43¹¹+29¹¹+23¹¹*2+22¹¹+14¹¹+12¹¹+7¹¹+1¹¹ (Luigi Morelli, 07/20/1999)
(11,2,16) 69¹¹+5¹¹=61¹¹+60¹¹+59¹¹*2+55¹¹*2+42¹¹+33¹¹*2+23¹¹+19¹¹+11¹¹+10¹¹+9¹¹+6¹¹+1¹¹ (Luigi Morelli, 07/27/1999)
(11,3,14) 72¹¹+56¹¹+12¹¹=69¹¹+66¹¹+53¹¹+49¹¹+43¹¹+34¹¹*2+20¹¹+9¹¹*2+8¹¹+4¹¹*2+2¹¹ (Nuutti Kuosa, 09/12/1999)
(11,5,13) 52¹¹+30¹¹+26¹¹+24¹¹+16¹¹=50¹¹+47¹¹+36¹¹+33¹¹+25¹¹+20¹¹+18¹¹+12¹¹+11¹¹+8¹¹*2+7¹¹+5¹¹ (Nuutti Kuosa, 08/31/1999)
(11,6,10) 46¹¹+39¹¹+37¹¹+22¹¹+4¹¹*2=43¹¹*2+41¹¹+31¹¹+28¹¹+26¹¹+23¹¹+18¹¹+16¹¹+15¹¹ (Nuutti Kuosa, 08/31/1999)
(11,7,9) 48¹¹+28¹¹+20¹¹+15¹¹*2+14¹¹+1¹¹=44¹¹+43¹¹+42¹¹+36¹¹*2+30¹¹+22¹¹+17¹¹+3¹¹ (Nuutti Kuosa, 09/20/1999)
(11,8,8) 67¹¹+52¹¹+51¹¹*2+39¹¹+38¹¹+35¹¹+27¹¹=66¹¹+60¹¹+47¹¹+36¹¹+32¹¹+30¹¹+16¹¹+7¹¹ (Nuutti Kuosa, 08/09/1999)
(12,9,9) 43¹²+33¹²+31¹²+23¹²+20¹²+16¹²*2+12¹²+6¹²=41¹²+40¹²+32¹²+30¹²*3+29¹²+5¹²*2 (Nuutti Kuosa, 08/12/1999)
(13,1,36) 990¹³=989¹³+709¹³+423¹³+305¹³+225¹³+157¹³+119¹³+98¹³+73¹³+52¹³+45¹³+37¹³*2+36¹³*2+31¹³+30¹³+28¹³+27¹³+25¹³+24¹³*4+22¹³*2+20¹³+16¹³+15¹³*2+9¹³+7¹³*2+3¹³*3 (Kevin O'Hare, 11/02/1999)
(13,5,31) 3327¹³+115¹³+47¹³+32¹³+27¹³=3291¹³+2840¹³+2180¹³+1323¹³+644¹³+443¹³+303¹³+211¹³+157¹³+88¹³+65¹³+38¹³+20¹³+15¹³+14¹³+12¹³*2+10¹³*2+9¹³*2+6¹³*3+5¹³*2+4¹³*3+2¹³*2 (Joe Crump, 10/20/1999)
(13,7,28) 77¹³+74¹³+61¹³+33¹³+28¹³+26¹³+22¹³=80¹³+46¹³+39¹³+16¹³*2+13¹³+10¹³+9¹³+8¹³*3+6¹³+5¹³*3+4¹³*8+3¹³*5 (Joe Crump, 10/03/1999)
(13,8,25) 256¹³+200¹³+189¹³+75¹³+60¹³+37¹³+32¹³+21¹³=257¹³+176¹³+125¹³+100¹³+47¹³+27¹³+24¹³+17¹³+15¹³*2+11¹³+10¹³+7¹³+6¹³+5¹³*2+3¹³*5+2¹³*4 (Joe Crump, 10/05/1999)
(13,9,22) 5556¹³+366¹³+263¹³+192¹³+141¹³+33¹³+15¹³+13¹³+10¹³=5416¹³+4898¹³+4609¹³+2290¹³+1347¹³+843¹³+542¹³+96¹³+47¹³+40¹³+26¹³+22¹³+20¹³+12¹³+8¹³+6¹³*2+3¹³+2¹³*4 (Joe Crump, 10/09/1999)
(13,10,22) 1717¹³+971¹³+579¹³+57¹³+24¹³+22¹³+16¹³+4¹³+3¹³+1¹³=1600¹³+1508¹³+1605¹³+404¹³+288¹³+193¹³+137¹³+108¹³+69¹³+46¹³+39¹³+33¹³+19¹³+14¹³+13¹³+11¹³+9¹³+8¹³+6¹³*3+2¹³ (Joe Crump, 10/08/1999)
(13,11,11) 31¹³+27¹³+23¹³*2+21¹³+19¹³+10¹³*3+6¹³*2=29¹³*2+28¹³+25¹³+24¹³+22¹³+15¹³+5¹³*2+3¹³+1¹³ (Nuutti Kuosa, 06/07/1999)
(14,1,40) 38¹⁴=35¹⁴*2+31¹⁴*4+30¹⁴*2+29¹⁴+28¹⁴+26¹⁴*4+25¹⁴*2+22¹⁴+21¹⁴*2+20¹⁴+18¹⁴*2+17¹⁴+14¹⁴*2+13¹⁴*3+12¹⁴*2+9¹⁴+8¹⁴+4¹⁴*7+2¹⁴ (Frank Clowes, 10/20/1999)
(14,9,31) 965¹⁴+856¹⁴+823¹⁴+582¹⁴+373¹⁴+207¹⁴+159¹⁴+89¹⁴+68¹⁴=983¹⁴+277¹⁴+54¹⁴+32¹⁴+23¹⁴+15¹⁴+14¹⁴+13¹⁴+12¹⁴+10¹⁴+9¹⁴*4+6¹⁴*3+4¹⁴+3¹⁴*12+1¹⁴ (Joe Crump, 09/29/1999)
(14,11,21) 1053¹⁴+885¹⁴+728¹⁴+707¹⁴+132¹⁴+97¹⁴+27¹⁴+25¹⁴+20¹⁴+16¹⁴+14¹⁴=1060¹⁴+497¹⁴+337¹⁴+246¹⁴+186¹⁴+68¹⁴+57¹⁴+45¹⁴+33¹⁴+22¹⁴+18¹⁴+9¹⁴*2+7¹⁴+4¹⁴+3¹⁴*6 (Joe Crump, 10/14/1999)
(14,14,14) 25¹⁴+20¹⁴*4+19¹⁴*2+18¹⁴*2+12¹⁴+9¹⁴+8¹⁴*2+1¹⁴=24¹⁴+22¹⁴*4+16¹⁴*2+15¹⁴*3+11¹⁴*2+2¹⁴*2 (Kevin O'Hare, 07/14/1999)
(16,24,25) 13917¹⁶+5278¹⁶+2949¹⁶+926¹⁶+639¹⁶+220¹⁶+165¹⁶+112¹⁶+57¹⁶+33¹⁶+24¹⁶+20¹⁶+19¹⁶+18¹⁶+15¹⁶*2+13¹⁶+12¹⁶*3+10¹⁶+2¹⁶*2+1¹⁶=13911¹⁶+10176¹⁶+8173¹⁶+1967¹⁶+1301¹⁶+424¹⁶+278¹⁶+134¹⁶+92¹⁶+76¹⁶+42¹⁶+29¹⁶+17¹⁶+7¹⁶*3+6¹⁶*5+5¹⁶*2+4¹⁶*2 (Joe Crump, 10/03/1999)
(17,1,71) 30¹⁷=28¹⁷*2+26¹⁷+25¹⁷*6+22¹⁷+21¹⁷*5+20¹⁷*2+19¹⁷*8+17¹⁷*8+16¹⁷*3+15¹⁷*4+14¹⁷*5+13¹⁷*4+11¹⁷*8+7¹⁷+6¹⁷+5¹⁷+4¹⁷+3¹⁷*4+2¹⁷*5+1¹⁷ (Larry Hays, 10/21/1999)
(19,8,106) 1140¹⁹+632¹⁹+154¹⁹+132¹⁹+114¹⁹+56¹⁹+50¹⁹+15¹⁹=1106¹⁹+791¹⁹+1091¹⁹+832¹⁹+486¹⁹+395¹⁹+288¹⁹+240¹⁹+200¹⁹+84¹⁹+71¹⁹+64¹⁹+45¹⁹+30¹⁹+27¹⁹+24¹⁹+21¹⁹+20¹⁹+19¹⁹+18¹⁹+13¹⁹+12¹⁹+11¹⁹*5+9¹⁹*2+7¹⁹*6+6¹⁹*3+5¹⁹*19+4¹⁹*40+3¹⁹+2¹⁹*7+1¹⁹ (Joe Crump, 10/21/1999)

Last update: 27th January 2000