Computing Minimal Equal Sums Of Like Powers 
Let's find a power of 6 equal to 6 powers of 6.
Join EulerNet !

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. b_{1 }>= b_{2}>= ... >= b_{n} a_{1 }> 1 m <= n 14132^{5}+ 220^{5 }= 14068^{5}+ 6237^{5}+ 5027^{5} 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. 
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 !
Older results can be found here
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: JeanCharles 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 
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) 
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 
© 19992000 JeanCharles Meyrignac <euler@free.fr> 