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. b1 >= b2>= ... >= bn a1 > 1 m <= n 141325+ 2205 = 140685+ 62375+ 50275 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: Jean-Charles 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 |
© 1999-2000 Jean-Charles Meyrignac <euler@free.fr> |