Computing Minimal Equal Sums Of Like Powers

Let's find a power of 6 equal to 6 powers of 6.
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Last updated
Index 08/17/2000
FAQ 08/17/2000
Top Producers 08/17/2000
Download 08/01/2000
Database 08/17/2000
Hall of Fame 08/12/2000
Records 08/12/2000
Progress status 08/16/2000
Most wanted 08/15/2000
Identities 02/26/2000
Links 08/11/2000
Resta 08/14/2000
Old Results 08/11/2000
Project details 07/14/2000
How it works 04/02/2000
Greetings 04/02/2000
AVL 04/02/2000
Scher 04/02/2000
NewsLetter 1 03/24/1999
NewsLetter 2 06/10/1999
Morpion Solitaire 07/16/2000
This project is dedicated to all those who are fascinated by powers and integers.

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

a1k+ a2k+ ... + amk = b1k+ b2k+ ... + bnk
with:
a1 >= a2 >= ... >= am
b1 >= b2>= ... >= bn
a1 > 1
m <= n
for example:
236+ 156+ 106= 226+ 196+ 36
141325+ 2205 = 140685+ 62375+ 50275

Lander, Parkin and Selfridge conjectured in 1966 that:

for every k > 3, m + n >= k

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 .
Check the EulerNet Top Producers here

19th August 2000: Mark Dodrill found 2 new solutions to (6,1,7) ! We are approaching 200,000 with EulerNet. Soon, we will switch to (6,1,6) to speed up the computations.
17th August 2000: Greg Childers found a new (6,1,7). The new Top Producers also revealed TWO older (6,1,7) -they were both found by Laurent Lucas- !
16th August 2000: Larry Hays found (17,12,36) and Laurent Lucas found (19,3,94) and (19,6,77)
15th August 2000: Greg Childers found a new solution to (6,1,7)
14th August 2000: Crazy day ! The Top Producers page has changed slightly, Jo MacLean found (23,1,434) and (31,1,11265). Eric Bainville sent me some very interesting pictures I'll put on the site soon, and Scott Chase sent solutions to (8,1,8), (8,3,5), (10,1,13), (10,4,11) and (10,5,9) !!! In fact, he seems to have discovered independently all our results on powers below 11 ten years ago, WOW !

Now, we know that eight eighth powers can sum to zero !!!
9668+5398+818-9548-7258-4818-3108-1588=0
11th August 2000: Kjeld Elholm Kristensen found (31,1,13147). JC Meyrignac found (20,1,160)
10th August 2000: Laurent Lucas found (13,8,14), (13,9,15), (13,10,14) and (14,11,19). JC Meyrignac found (19,3,103)
9th August 2000: Jean-Charles Meyrignac found (13,7,14) and (14,7,19)
8th August 2000: Jean-Charles Meyrignac found (13,6,15) and Kjeld Elholm Kristensen found (31,1,15063) !
7th August 2000: Jean-Charles Meyrignac found (13,5,27) and (13,6,23). Laurent Lucas found a new solution to (6,1,7)
5th August 2000: Laurent Lucas found (14,8,20) and (14,12,22), Torbjörn Alm found (22,4,145), Maurice Blondot found a new solution to (7,2,6) and Greg Childers found two new solutions to (6,1,7). Also added the names of the records holders on the lowest number of terms and changed the design of the tables a little bit.
1st August 2000: Torbjörn Alm found (17,5,58) and (18,2,81). I would like to apologize to Joe and John Michael Crump. Their records are still out of the table !
30th July 2000: Torbjörn Alm found (19,4,88) and (20,4,112), also he suggested that every new record should be marked with the initials of the discoverer. Excellent idea !
28th July 2000: Laurent Lucas found (25,4,524)
25th July 2000: Laurent Lucas found (18,6,64) and (18,9,46). Also added a FAQ if you want to join the search on (6,1,7)
24th July 2000: Jean-Charles Meyrignac found (13,3,27)
23rd July 2000: Version 4.17 is available for download here

Older results can be found here

KNOWN LOWER BOUNDS
k\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 2                          
3 3 2                        
4 3 2                        
5 4
(LP)		 3
(BS)		                        
6 7
(LP)		 5
(EB,GR)		 3                      
7 7
(MD)		 6
(JCM)		 5
(RE,RS)		 4
(RE)		                    
8 8
(SC)		 8
(RE)		 5
(SC)		 5
(RE)		                    
9 12
(JCM)		 10
(LM)		 9
(RE,RS)		 6
(RE)		 5
(RE)		                  
10 13
(SC)		 12
(NK)		 12
(TN)		 11
(SC)		 9
(SC)		 6
(RE)		                
11 16
(LM)		 16
(LM)		 14
(NK)		 13
(AN)		 13
(NK)		 10
(NK)		 9
(NK)		 8
(NK)		            
12 26
(JCM)		 27
(TA)		 27
(JCM)		 22
(MW)		 21
(TA)		 18
(NK)		 17
(NK)		 17
(NK)		 9
(NK)		          
13 33
(JC)		 30
(TA)		 27
(JCM)		 28 27
(JCM)		 15
(JCM)		 14
(JCM)		 14
(LL)		 15
(LL)		 14
(LL)		 11
(NK)		      
14 40
(FC)		 35
(LL)		 33
(JCM)		 30
(TA)		 31
(LL)		 20
(LL)		 19
(JCM)		 20
(LL)		 21
(TA)		 20
(TA)		 19
(LL)		 22
(LL)		 22
(JC)		 14
(KO)
15 46
(NK)		 42
(DA)		 41
(LL)		 34
(JCM)		 35
(LL)		 36
(LL)		 35
(TA)		 29
(LL)		 24
(LL)		 25 26 27 28 29
16 77
(NK)		 76
(NH)		 76
(LHU)		 72
(LHU)		 73
(LL)		 74
(LHU)		 75
(LL)		 42
(LHU)		 43 44 45 46 47
(JC)		 31
(JC)
17 71
(LHA)		 69
(LL)		 62
(LHA)		 59
(LHA)		 58
(TA)		 53
(LHA)		 50
(LHA)		 51 52 53 54 36
(LHA)		 37 38
18 96
(NK)		 81
(TA)		 82
(LHA)		 81
(LHA)		 80
(LHA)		 64
(LL)		 65 66 46
(LL)		 47 48 49 50 51
19 119
(JCM)		 112
(LL)		 94
(LL)		 88
(TA)		 79
(LHA)		 77
(LL)		 78 79 80 81 82 83 84 85
20 160
(JCM)		 113
(LL)		 114 112
(TA)		 113 114 115 116 117 118 119 120 121 122
21 240
(JCM)		 180
(LL)		 181 156
(TA)		 132
(LHA)		 133 134 135 136 137 138 139 140 141
22 300
(JCM)		 234
(LL)		 235 145
(TA)		 146 147 148 149 150 151 152 153 154 155
23 434
(JML)		 420
(LL)		 257
(LL)		 243
(LL)		 244 245 246 247 248 249 250 251 252 253
24 485
(JCM)		 360
(LL)		 361 362 363 364 365 366 367 368 369 370 371 372
25 845
(LL)		 817
(LL)		 563
(LL)		 542
(LL)		 543 544 535
(LHA)		 536 537 538 539 540 541 542
26 869
(JML)		 870 871 872 873 874 875 876 877 878 879 880 881 882
27 1059
(JML)		 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072
28 1975
(JML)		 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988
29 3989
(JML)		 3990 3991 3992 3993 3994 3995 3996 3997 3998 3999 4000 4001 4002
30 4033
(LL)		 4034 4035 4036 4037 4038 4039 4040 4041 4042 4043 4044 4045 4046
31 11265
(JML)		 11266 11267 11268 11269 11270 11271 11272 11273 11274 11275 11276 11277 11278
32 6038 
(JML)		 6039 6040 6041 6042 6043 6044 6045 6046 6047 6048 6049 6050 6051

AN: Aleksi Niemelä BS: Bob Scher DA: David Alten EB: Edward Brisse EB2: Eric Bainville
FC: Frank Clowes GR: Giovanni Resta JC: Joe Crump JCM: Jean-Charles Meyrignac JMC: John Michael Crump
JML: Joe MacLean KO: Kevin O'Hare LHA: Larry Hays LHU: Luke Huitt LL: Laurent Lucas
LM: Luigi Morelli LP: Lander and Parkin MD: Mark Dodrill MW: Mac Wang NH: Norman Ho
NK: Nuutti Kuosa PG: Pascal Gelebart RE: Randy Ekl RS: Rizos Sakellariou TA: Torbjörn Alm
TN: Tommy Nolan KEK: Kjeld Elholm Kristensen SC: Scott Chase
LOWEST NUMBER OF TERMS
k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
m+n 3 3 4 4 5
(LP, BS)		 6 8
(MD,
JCM, RE, RS)		 8
(SC)		 10
(RE)		 12
(RE)		 16
(NK)		 18
(NK)		 21
(JCM)		 26
(LL,
JCM)		 33
(LL)		 45
(JC)
k 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
m+n 48
(LHA)		 55
(LL)		 83
(LL)		 115
(LL)		 137
(LHA)		 149
(TA)		 247
(LL)		 362
(LL)		 542
(LHA)		 465
(JMC)		 1060
(JML)		 1759
(JMC)		 1203
(JMC)		 4034
(LL)		 11235
(JML)		 6039
(JML)
LEGEND
power (k)
number of left terms (m)
number of right terms (n), unexplored
number of right terms (n), known lower bound
number of right terms (n), best lower bound currently found
number of right terms (n), best lower bound conjectured
number of right terms (n), best lower bound proved
number of right terms (n) that doesn't need to be explored

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© 1999-2000 Jean-Charles Meyrignac <euler@free.fr>