Computing Minimal Equal Sums Of Like Powers

Let's find a power of 6 equal to 6 powers of 6.
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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: 52 = 42+32 = 25 123+13 = 103+93 = 1729 1584+594 = 1344+1334 = 635318657 4224814 = 4145604+2175194+958004 = 31858749840007945920321 1445 = 1335+1105+845+275 = 61917364224 141325+2205 = 140685+62375+50275 = 563661204304422162432 236+156+106 = 226+196+36 = 160426514 9668+5398+818 = 9548+7258+4818+3108+1588 = 765381793634649192581218 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

15th of March 2020: Duncan Moore found a new record for 10th power: (10,5,6). Congratulations!
27th of December 2011: Massive update with Scott Chase's and Rolan Christofferson's results.
16th of September 2011: Scott Chase discovered a lot of new results, consult the database file. The most important results are (19,15,17) and (27,27,34) which provides new records for 19th and 27th powers.
11st of September 2011: Scott Chase discovered (17,5,27), (17,13,16),(17,11,20), (18,12,26), (22,20,22), (22,21,22) and (21,13,26) ! This gives a new record of 42 terms on 22th power.
4th of September 2011: Scott Chase discovered (18,13,15) and (17,13,17) ! The (18,13,15) reduces from 30 to 13+15=28 terms on 18th power.
27th of August 2011: Scott Chase discovered (16,7,26) !
25th of August 2011: Laurent Lucas retrieved an old hard-disk, and a lot of records, the most notable being (20,17,17) !
21st of August 2011: Scott I. Chase just discovered (16,9,22) !
The EulerNet project has finished recently to compute all solutions below 250000, and Robert Gerbicz submitted an article about in on Arxiv.org
28th of April 2010: I have both good news and bad news

The bad news is that the eulernet.org server has been impossible to reach this last month, so the clients should have an empty queue, waiting for some work.
Greg Childers, who maintains the server, is very busy, and I don't want to bother him too much, since we reached a plateau since a few months.

The good news are:

1. The Eulernet project reached its 11th birthday in January. It's a considerable amount of time for a distributed project.
2. A new distributed project about (6,2,5) has been started using the BOINC platform. The link is http://www.rechenkraft.net/yoyo/
   To join, you just have to use your BOINC account (or create a new one if you don't have one), then click on Your account, then change the yoyo@home preferences. Uncheck all projects, check Euler 625, then check "Run test applications", since the Euler625 project is still in testing... After that, install the BOINC client, and follow the instructions on the home page.

As long as the new project has not been finished, the EulerNet project will remain alive, and we'll try to collect all the reports, so that no participant will lose its credits.
I'll also check the solutions of the new project regularly and report the new solutions here.
Older results can be found here

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KNOWN LOWER BOUNDS

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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)

6 (DMO)

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)

35 (SIC)

29 (SIC)

30

31 (SIC)

23 (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)

31 (SIC)

28 (SIC)

29 (GC)

25 (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)

24 (SIC)

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

191 (SIC)

179 (SIC)

180

181

179 (TA)

175 (LL)

168 (GC)

167 (LL)

157 (GC)

158

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LOWEST NUMBER OF TERMS

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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)	11 (DMO)	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)	41 (SIC)	44 (LL)	48 (LL)	52 (SIC,LL)	52 (SIC)	61 (SIC)	63 (SIC)	67 (SIC)	69 (SIC)	72 (SIC)	75 (SIC)

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ACO: Andrea Concaro	AL: Aloril	AN: Aleksi Niemelä	BS: Bob Scher	DA: David Alten
DM: Douglas McNeil	DMO: Duncan Moore	EB: Edward Brisse	EB2: Eric Bainville	FC: Frank Clowes
FY: Fumitaka Yura	GC: Greg Childers	GR: Giovanni Resta	JC: Joe Crump	JCM: Jean-Charles 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: Subba-Rao	TA: Torbjörn Alm	TN: Tommy Nolan

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LEGEND

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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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Visitor number since 1st April 2000 (project started 1st January 1999).
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© 1999-2005 Jean-Charles Meyrignac <euler@free.fr>