Computing Minimal Equal Sums Of Like Powers
Let's find a power of 6 equal to 6 powers of 6.
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Index 04/28/2010
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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.
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

KNOWN LOWER BOUNDS
 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

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) 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)

 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

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