Computing Minimum Equal Sums Of Like Power

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	05/17/2009
FAQ	01/22/2002
Top Producers	Frequently !
Search this site	01/05/2003
Download	09/30/2002
Mailing list	09/19/2000
Solitaire Competition	12/31/2002
Peg Solitaire tough problems	01/13/2003
Database	05/17/2009
K10 search	02/22/2003
Old results	10/09/2002
Cheap software	01/30/2003
Links	08/21/2001
Identities	02/09/2001
Theorems	01/24/2003
Prime solutions	09/15/2005
Prime database	05/17/2009
High power solutions	04/10/2002
Records	02/14/2001
Progress status	03/25/2001
Most wanted	01/24/2003
Resta	11/03/2000
Project details	01/27/2001
Taxicab numbers	12/18/2002
How it works	04/02/2000
Greetings	01/04/2001
AVL	04/02/2000
Scher	12/28/2000
Morpion Solitaire	04/15/2003
Knight tours	04/25/2003
Peg Infos	12/30/2002
Javascript	10/24/2002
Dungeon Keeper 2	11/10/2006

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:

a₁^k+ a₂^k+ ... + a_m^k = b₁^k+ b₂^k+ ... + b_n^k with: a₁ >= a₂ >= ... >= a_m
b₁ >= b₂ >= ... >= b_n
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

10th August 2009: As you may have noticed, the stats page has not been updated since the 14th of July. The problem is due to the fact that I cannot access to the FTP server containing the daily stats.
Don't worry, nothing is lost, since ranges continue to be assigned, and all computations are logged. I apologize for the inconvenience. Greg Childers has been contacted to solve this problem.
14th June 2009: The EulerNet server has some problems.
Scott Chase found (16,10,22), forcing me to update the records. Andrea Concaro now officially holds the records on powers 31 and 32 ! (Sorry, the table is too small)
Tito Piezas published an impressive 200+ pages online book about identities: http://sites.google.com/site/tpiezas/Home.
If you want to see what I look like, here is a scan about the crosswords championship: http://cboyer.club.fr/MeyrignacChampion2008.pdf (PDF thanks to Christian Boyer).
If I have some courage, I'll write a short introduction about Equal Sums of Like Powers for Dummies, since a lot of people don't understand the small introduction.
17th May 2009: Scott Chase sent two new results : (12,4,14) and (16,10,23).
A project about computing solutions to (4,1,3) finished recently: http://euler413.narod.ru/ with the discovery of 4 new small solutions. You can get the source code here: http://robert.gerbicz.googlepages.com/
Christian Boyer created a site about Morpion Solitaire: http://www.morpionsolitaire.com/, where he keeps track of the best records.
Uwe Hollerbach confirmed that TaxiCab(6) = 24153319581254312065344 in March 2008. The current best results of TaxiCab and CabTaxi are here: http://cboyer.club.fr/Taxicab.htm
In May 2008, Uwe confirmed that CabTaxi(10) = 933528127886302221000.
In November 2006, there was an article about Nuuti's discovery of (8,4,4): http://www.maa.org/editorial/mathgames/mathgames_11_13_06.html
In June 2008, I (Jean-Charles Meyrignac) became the french champion of Mots Fléchés (swedish grids style).
If you are interested in magic squares and sums of powers, I recommend that you take a look at Christian's page: http://www.multimagie.com/English/SquaresOfCubes.htm
Peter J. Ansell maintains a page about sixth power: http://www.computer-man.demon.co.uk/
I finally updated the database.txt and dataprime.txt files !
Rolan Christofferson maintains a list of (6,1,7) solutions as a Google spreadsheet: https://spreadsheets.google.com/ccc?key=pNPrFHlUozfQnE1_mCVdZVQ

31th January 2009: Sorry for the lack of recent since the last years. The project is now in its tenth year of computation !
A big thanks to all participants, let's hope that our computations will prove to be useful in a near future.
I recently got some questions about the program, mainly to fix the problems. I'm very sorry to not react faster, mostly because I'm working on other projects now, but I don't think I can help a lot on these problems.
Without Greg Childers' constant support for all these years, the project would have stopped a long time ago.
The last years, the records didn't progress well (but I probably forgot a mail, so send me your results if you found something new).
Today, Scott Chase discovered two new results on the twelveth power: (12,2,16) and (12,5,14).
9th November 2006: Nuutti Kuosa found (8,4,4), which is a new record of the form (k,m,k-m) ! Here is his splendid discovery:

3113⁸+2012⁸+1953⁸+861⁸=2823⁸+2767⁸+2557⁸+1128⁸

8th November 2006: Robert Gerbicz found 2 new solutions for (4,1,3), using his new program. His program is 100 times faster than Daniel J. Bernstein's !!!
6th November 2006: Nuutti Kuosa sent a lot of new solutions for powers 7 and 8, but no record.
5th November 2006: Sorry for the lack of update, but I was busy with the contest site I code and also on a Flash version of my peg solitaire game. I was also very lazy.
4th November 2006: Jaroslaw Wroblewski found the following 1,3,7 multigrades:

(7,4,4) 33704+32317+14977+9803=33450+32630+14664+10057
(7,4,4) 76925+52473+50279+15187=74895+64419+27857+27693
(7,4,4) 58711+42312+38285+9544=56170+51782+24427+16473
(7,3,5) 1205011+744503+678161=1186971+921065+235693+187391+96555

2nd November 2006: Robert Gerbicz found a new solution to (4,1,3):

117112081⁴=87865617⁴+34918520⁴+106161120⁴

25th October 2006: Robert Gerbicz sent me a faster program to compute (6,2,5), in other words, it may be able to boost our current distributed project !
The source code is here: euler.c
9th October 2006: Duncan Moore found several new results on 8th power:

1056⁸+129⁸+1⁸=1017⁸+873⁸+710⁸+280⁸+170⁸+86⁸
469⁸+388⁸+314⁸+2⁸=482⁸+274⁸+210⁸+149⁸+34⁸
564⁸+139⁸+52⁸+8⁸=540⁸+476⁸+364⁸+299⁸+174⁸
908⁸+530⁸+457⁸+349⁸=850⁸+791⁸+678⁸+125⁸+46⁸

3rd October 2006: Jaroslaw Wroblewski found a 18th solution to (9,k,10-k):

2152⁹+979⁹+752⁹+514⁹=2101⁹+1793⁹+1021⁹+870⁹+750⁹+112⁹

20th September 2006: I migrated my DNS from Gandi to 1and1. It's much cheaper and provides the same service.
11th September 2006: Alexander Fischer improved the non-crossing knight tours of the 14x14 from 134 to 135, and 16x16 from 182 to 183.
3rd August 2006: Jaroslaw Wroblewski collected solutions to 3rd and 4th powers:
http://www.math.uni.wroc.pl/~jwr/eslp/tables.htm
13th March 2006: Seiji Tomita discovered several new solution to (4,1,3):

582665296⁴+260052385⁴+186668000⁴=589845921⁴
1670617271⁴+632671960⁴+50237800⁴=1679142729⁴
22495595284040⁴+7592431981391⁴+27239791692640⁴=29999857938609⁴

and the Ramanujan-type identity:

(2x²+36x-54)⁴+(2x²-36x-54)⁴+(4x²-108)⁴+(4x²+108)⁴+(3x²+81)⁴=(5x²+135)⁴

His homepage is http://www3.alpha-net.ne.jp/users/fermat/index.html
16th January 2006: Doomeva from AMDUsers team sent me several logos:

11th January 2006: Tito Piezas posted a new conjecture (and kindly dedicated it to us):
http://www.geocities.com/titus_piezas/Timeline1.htm
4th January 2006: Jaroslaw Wroblewski built a list of 40 solutions to a⁴-b⁴=c⁴-d⁴=e⁴-f⁴.
1st January 2006: Yasutoshi Kohmoto sent a nice equation for 2006:

2006⁶=3858364³+1976500³+24308³=3590596³+2662272³-25512³=3561535³+2713705³+26786³

And yes, it's the seventh birthday of our distributed project !
Older results can be found here

KNOWN LOWER BOUNDS

k\m

3
(RF)

4
(LP)

3
(BS)

7
(LP)

5
(EB&GR)

3
(SR)

7
(MD)

6
(JCM)

5
(RE)

4
(RE)

8
(SIC)

7
(SIC)

5
(SIC)

4
(NK)

10
(JW)

8
(JW)

8
(JW)

6
(RE)

5
(RE)

12
(JW)

12
(NK)

11
(JW)

9
(JW)

7
(JW)

6
(RE)

15
(JW)

14
(JW)

11
(NK)

10
(NK)

9
(NK)

8
(NK)

7
(NK)

19
(JW)

16
(SIC)

15
(JW)

14
(SIC)

11
(JW)

7
(GC)

21
(JW)

20
(JW)

18
(JW)

15
(JW)

15
(JCM)

13
(FY)

9
(GC)

25
(JW)

21
(JW)

19
(JW)

18
(JW)

17
(JW)

16
(JW)

13
(GC)

12
(JW)

9
(JW)

28
(JW)

24
(JW)

23
(JW)

19
(JW)

16
(JW)

14
(JW)

14
(JW)

13
(JW)

11
(JW)

49
(JW)

36
(JW)

35
(JW)

27
(SIC)

22
(GC)

23
(SIC)

22
(SIC)

12
(TA)

39
(JW)

35
(JW)

33
(JW)

30
(GC)

23
(TA)

24
(GC)

23
(GC)

20
(DM)

21
(TA)

18
(GC)

14
(GC)

57
(JW)

44
(FY)

43
(TA)

34
(TA)

36
(TA)

35
(MLI)

34
(FY)

29
(GC)

28
(FY)

29
(FY)

16
(GC)

15
(GC)

51
(JW)

43
(GC)

40
(FY)

41
(FY)

36
(FY)

33
(GC)

29
(GC)

31
(GC)

27
(GC)

21
(GC)

17
(GC)

61
(JW)

61
(FY)

60
(FY)

59
(FY)

53
(TA)

39
(TA)

41
(GC)

35
(FY)

35
(GC)

35
(FY)

26
(GC)

75
(FY)

72
(FY)

67
(TA)

60
(FY)

57
(TA)

50
(FY)

46
(TA)

41
(GC)

39
(GC)

37
(GC)

34
(GC)

31
(DM)

27
(GC)

25
(TA)

95
(TA)

75
(FY)

72
(FY)

73
(FY)

59
(TA)

57
(FY)

54
(FY)

52
(TA)

36
(GC)

37
(GC)

36
(GC)

37
(GC)

105
(FY)

91
(FY)

87
(FY)

86
(FY)

81
(FY)

72
(FY)

62
(FY)

53
(GC)

49
(FY)

49
(GC)

41
(TA)

36
(GC)

37
(GC)

33
(GC)

124
(FY)

116
(FY)

109
(FY)

97
(FY)

85
(TA)

85
(FY)

71
(FY)

69
(GC)

57
(FY)

58
(FY)

55
(GC)

52
(FY)

137
(FY)

118
(FY)

104
(FY)

94
(GC)

89
(GC)

81
(GC)

80
(GC)

76
(GC)

70
(GC)

72
(LL)

69
(DM)

67
(GC)

155
(FY)

136
(FY)

119
(FY)

116
(GC)

111
(FY)

106
(FY)

99
(FY)

100

81
(GC)

75
(TA)

72
(LL)

70
(TA)

52
(SMS&JW)

54
(LL)

163
(FY)

146
(TA)

132
(FY)

118
(FY)

119

104
(FC)

99
(FC)

100

87
(GC)

63
(GC)

56
(GC)

204
(FY)

147
(FY)

148
(JCM)

128
(FY)

129

124
(JCM)

124
(TA)

124
(FY)

99
(FY)

96
(GC)

97
(JCM)

98
(GC)

74
(GC)

73
(GC)

74
(FY)

173
(TA)

168
(FC)

145
(FY)

146

147

142
(FY)

143

134
(FY)

125
(FY)

120
(FY)

115
(FC)

102
(GC)

96
(GC)

89
(GC)

84
(FY)

191
(FY)

188
(FY)

189

164
(FY)

139
(FY)

140

141

142

135
(FC)

134
(FC)

135
(GC)

106
(FY)

107

108

109

80
(TA)

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)

230
(FY)

214
(FY)

210
(TA)

210
(FY)

194
(FY)

195

196

184
(FY)

183
(FY)

179
(TA)

178
(TA)

168
(GC)

169

157
(GC)

158

LOWEST NUMBER OF TERMS

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	30 (GC)	33 (GC)	37 (TA)	40 (TA,GC)	44 (GC)	46 (GC)	48 (LL)	54 (LL)	54 (TA)	63 (LL,TA)	66 (GC)	69 (TA)	72 (GC)	77 (ACO,GC)	83 (ACO)

ACO: Andrea Concaro	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: 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

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

Visitor number Visitor

since 1st April 2000 (project started 1st January 1999).
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