Computing Minimal Equal Sums Of Like Powers 

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: b_{1} >= b_{2} >= ... >= b_{n} a_{1} > 1 m <= n 12^{3}+1^{3} = 10^{3}+9^{3} = 1729 158^{4}+59^{4} = 134^{4}+133^{4} = 635318657 422481^{4} = 414560^{4}+217519^{4}+95800^{4} = 31858749840007945920321 144^{5} = 133^{5}+110^{5}+84^{5}+27^{5} = 61917364224 14132^{5}+220^{5} = 14068^{5}+6237^{5}+5027^{5} = 563661204304422162432 23^{6}+15^{6}+10^{6} = 22^{6}+19^{6}+3^{6} = 160426514 966^{8}+539^{8}+81^{8} = 954^{8}+725^{8}+481^{8}+310^{8}+158^{8} = 765381793634649192581218 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. Check the EulerNet Top Producers here 
k\m  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16 
2  2  
3  3  2  
4  3
(RF) 
2  
5  4
(LP) 
3
(BS) 

6  7
(LP) 
5
(EB&GR) 
3
(SR) 

7  7
(MD) 
6
(JCM) 
5
(RE) 
4
(RE) 

8  8
(SIC) 
7
(SIC) 
5
(SIC) 
4
(NK) 

9  10
(JW) 
8
(JW) 
8
(JW) 
6
(RE) 
5
(RE) 

10  12
(JW) 
12
(NK) 
11
(JW) 
9
(JW) 
7
(JW) 
6
(RE) 

11  15
(JW) 
14
(JW) 
11
(NK) 
10
(NK) 
9
(NK) 
8
(NK) 
7
(NK) 

12  19
(JW) 
16
(SIC) 
15
(JW) 
14
(SIC) 
14
(SIC) 
11
(JW) 
7
(GC) 

13  21
(JW) 
20
(JW) 
18
(JW) 
18
(JW) 
15
(JW) 
15
(JCM) 
13
(FY) 
9
(GC) 

14  25
(JW) 
21
(JW) 
19
(JW) 
18
(JW) 
17
(JW) 
16
(JW) 
13
(GC) 
12
(JW) 
9
(JW) 

15  28
(JW) 
24
(JW) 
23
(JW) 
19
(JW) 
20  16
(JW) 
14
(JW) 
14
(JW) 
13
(JW) 
11
(JW) 

16  49
(JW) 
36
(JW) 
36
(JW) 
35
(JW) 
35
(JW) 
35
(JW) 
26
(SIC) 
22
(GC) 
22
(SIC) 
22
(SIC) 
12
(TA) 

17  39
(JW) 
40  35
(JW) 
33
(JW) 
27
(SIC) 
28  23
(TA) 
24
(GC) 
23
(GC) 
20
(DM) 
20
(SIC) 
18
(GC) 
16
(SIC) 
14
(GC) 

18  57
(JW) 
57
(JW) 
57
(JW) 
44
(FY) 
43
(TA) 
34
(TA) 
35  36
(TA) 
35
(MLI) 
34
(FY) 
29
(GC) 
26
(SIC) 
15
(SIC) 
16
(GC) 

19  51
(JW) 
52  53  51
(JW) 
43
(GC) 
40
(FY) 
41
(FY) 
36
(FY) 
33
(GC) 
29
(GC) 
30  31
(GC) 
23
(SIC) 
24  17
(SIC) 
17
(GC) 
20  61
(JW) 
62  61
(FY) 
60
(FY) 
59
(FY) 
53
(TA) 
39
(TA) 
40  35
(LL) 
35
(FY) 
35
(FY) 
35
(FY) 
35
(GC) 
35
(FY) 
35
(FY) 
26
(GC) 
21  75
(FY) 
72
(FY) 
67
(TA) 
60
(FY) 
57
(TA) 
50
(FY) 
51  46
(TA) 
41
(GC) 
39
(GC) 
37
(GC) 
34
(GC) 
26
(SIC) 
25
(LL) 
25
(TA) 
26 
22  95
(TA) 
75
(FY) 
76  72
(FY) 
73  73
(FY) 
59
(TA) 
57
(FY) 
54
(FY) 
54
(FY) 
52
(TA) 
36
(GC) 
37  37
(GC) 
36
(GC) 
37
(GC) 
23  105
(FY) 
91
(FY) 
87
(FY) 
86
(FY) 
81
(FY) 
72
(FY) 
62
(FY) 
63  62
(FY) 
53
(GC) 
49
(FY) 
47
(LL) 
41
(TA) 
36
(GC) 
37
(GC) 
33
(GC) 
24  124
(FY) 
116
(FY) 
109
(FY) 
97
(FY) 
85
(TA) 
85
(FY) 
71
(FY) 
72  65
(LL) 
66  57
(FY) 
58  58
(FY) 
55
(GC) 
52
(FY) 
53 
25  137
(FY) 
118
(FY) 
104
(FY) 
94
(GC) 
89
(GC) 
90  81
(GC) 
82  80
(GC) 
80
(GC) 
74
(SIC) 
70
(GC) 
68
(SIC) 
69  69
(DM) 
67
(GC) 
26  155
(FY) 
133
(LL) 
119
(FY) 
116
(GC) 
111
(FY) 
106
(FY) 
99
(FY) 
100  81
(GC) 
82  75
(TA) 
72
(LL) 
70
(TA) 
52
(SMS&JW) 
53  54
(LL) 
27  162
(LL) 
146
(TA) 
132
(FY) 
118
(FY) 
119  104
(FC) 
99
(FC) 
100  87
(GC) 
88  89  63
(GC) 
64  65  66  56
(GC) 
28  183
(LL) 
147
(FY) 
148
(JCM) 
128
(FY) 
129  124
(JCM) 
124
(TA) 
124
(FY) 
96
(LL) 
96
(LL) 
96
(GC) 
97
(JCM) 
98
(GC) 
74
(GC) 
73
(GC) 
74
(FY) 
29  173
(TA) 
168
(FC) 
145
(FY) 
146  147  136
(LL) 
137  119
(LL) 
120  108
(LL) 
109  102
(GC) 
96
(GC) 
89
(GC) 
90  84
(FY) 
30  191
(FY) 
188
(FY) 
189  164
(FY) 
139
(FY) 
140  141  142  133
(LL) 
134
(FC) 
135
(GC) 
106
(FY) 
107  108
(LL) 
109
(LL) 
80
(TA) 
31  211
(GC) 
200
(GC) 
199
(GC) 
191
(GC&JW) 
184
(GC&JW) 
182
(GC&JW) 
160
(TA&JW) 
161  162  162
(GC&JW) 
153
(TA&JW) 
151
(TA&JW) 
149
(TA&JW) 
150  143
(TA&JW) 
127
(TA&JW) 
32  230
(FY) 
214
(FY) 
210
(TA) 
210
(FY) 
194
(FY) 
195  196  184
(FY) 
183
(FY) 
183
(FY) 
179
(TA) 
175
(LL) 
168
(GC) 
167
(LL) 
157
(GC) 
158 
k  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17 
m+n  3 
4 
4 
5 (LP,BS) 
6 (SR) 
8 (RE,MD,JCM) 
8 (NK,SIC) 
10 (RE,JW) 
12 (RE,JW) 
14 (NK) 
14 (GC) 
17 (GC) 
18 (JW) 
21 (JW) 
23 (TA) 
28 (GC) 
k  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32 
m+n  28 (SIC) 
32 (SIC) 
34 (LL) 
39 (SIC,LL) 
42 (SIC) 
44 (LL) 
48 (LL) 
52 (SIC,LL) 
52 (SIC) 
61 (SIC) 
63 (SIC) 
67 (SIC) 
70 (SIC,LL) 
77 (LL,GC) 
75 (SIC) 
AL: Aloril  AN: Aleksi Niemelä  BS: Bob Scher  DA: David Alten  DM: Douglas McNeil 
EB: Edward Brisse  EB2: Eric Bainville  FC: Frank Clowes  FY: Fumitaka Yura  GC: Greg Childers 
GR: Giovanni Resta  JC: Joe Crump  JCM: JeanCharles Meyrignac  JMC: John Michael Crump  JML: Joe MacLean 
JW: Jaroslaw Wroblewski  KEK: Kjeld Elholm Kristensen  KO: Kevin O'Hare  LHA: Larry Hays  LHU: Luke Huitt 
LL: Laurent Lucas  LM: Luigi Morelli  LP: Lander and Parkin  MD: Mark Dodrill  ML: Marcin Lipinski 
MW: Mac Wang  NH: Norman Ho  NK: Nuutti Kuosa  PG: Pascal Gelebart  RE: Randy Ekl 
RF: Roger Frye  RS: Rizos Sakellariou  SIC: Scott I.Chase  SR: SubbaRao  TA: Torbjörn Alm 
TN: Tommy Nolan 
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 
© 19992005 JeanCharles Meyrignac <euler@free.fr> 