# Problem of the Month (June 2008)

This month we investigate four problems involving powers:

1. Most recreational math enthusiasts know that 153 is a narcissistic number because 13 + 53 + 33 = 153. In other words, if n{k} stands for the sum of the kth powers of the digits of n, we have 153{3} = 153. A full list of the 88 narcissistic numbers can be found here.

A generalization of narcissistic numbers are recurring digital invariants, or RDI's. For example 55{3} = 250, 250{3} = 133, and 133{3} = 55, which we abbreviate 55{3,3,3} = 55. These are simply cycles rather than fixed points of this map. A partial list of RDI's can be found here.

We are interested in a further generalization of this idea. For a given positive integer n, what is the smallest list of powers for which n{n1, n2, n3, ... nk} = n? For example, 18{1,3,1} = 18 because 18{1} = 9, 9{3} = 729, and 729{1} = 18. We call such numbers generalized recurring digital invariants, or GRDI's. Of course smallest could mean shortest, smallest sum, or smallest maximum element.

What are the best results for large n? How about n=2008? What numbers are GRDI's?

2. How close can a sum of the form Σ ± aibi be to 0 if the ai and bi are the numbers from 1–2n? What are the minimal differences for large n? Can the difference ever be exactly 0? If not, how fast do these minimal differences grow?

3. For k>1 and n>1, what is the smallest collection of k different nth powers that have exactly the same collection of digits? For example, for k=2 and n=3, we have 53 = 125 and 83 = 512. How about larger values of k and n?

4. The equation 24 = 42 is the only non-trivial integer equation of xy = yx. But if we are allowed to have more than one term on each side, there are other true equations where the bases and exponents have been switched, including: 23 + 25 + 35 + 43 + 62 + 72 = 32 + 52 + 53 + 34 + 26 + 27. Can you find some solutions with fewer terms?

1. Here are the smallest lists for all the 2-digit numbers:

Smallest GRDI Exponents
nmin lengthmin summin max
10none
11{1,7,1}{1,2,2,2,2,2,1,2,2,2,1}
12{7,1}{2,2,1,3,3,2,1}
13{1,4,1}{1,2,2,2,1}
14{4,1}{2,1,2,3,2,1}
15{2,4,2}{1,2,2,3,2,3,2,1}
16{2,4,1}{1,2,2,1}
17{3,2,2}{2,2,2,2,2,1}
18{1,3,1}
19{4,1}{2,3,3,2}
20{13,1}{2,4,4,1}{2,2,2,2,2,2,2,2}
21{13,1}{2,2,1,4,2}{2,2,2,2,3,3,3,3,2,3,2}
22{13,1}{1,7,1}{3,2,2,2,3,2}
23{1,7,1}{4,1,4,4,3,2}
24{5,7,1}{1,3,5,1}{3,2,3,3,2,3,2}
25{7,1}{2,4,1}{2,2,2,1,2,2}
26{9,1}{2,5,1}{1,3,3,2}
27{5,1}{2,3,2,3}
28{7,1}{3,2,2,1,3}
29{4,1,2}{2,2,1,2,2,2}
30{8,5,1}{3,2,2,3}
31{1,10,1}{1,4,3,4,1}
32{11,1}{2,4,4,2,4}
33{4,3,2}{1,3,3,2}
34{14,1}{1,3,2}
35{9,1}{3,2,3,3,2}
36{8,1}{1,2,2,2,2,2,2,2,2,1,2}
37{4,1,2}{2,1,1,2,2}
38{3,11,1}{2,2,6,1}{3,2,3,3,2}
39{3,13,1}{1,2,5,2}{3,4,4,4,3,2}
40{13,1}{5,4,2,2}{2,3,2,2,3,3,2,2}
41{13,1}{2,3,2}{1,2,2,2,2,2,2,1,2,2,2}
42{5,13,1}{3,1,4,2,2,2}{2,2,2,2,2,2,2,2}
43{17,1}{1,10,1}{1,3,4,4,3,4,2}
44{3,15,1}{1,7,7,1}{3,2,2,4,4,4,4,4,2,2}
45{17,1}{1,7,1}{2,1,2,2,2,2,2,2,1,2,2}
46{12,1}{2,3,4,2}{2,1,2,3,3,2,2}
47{12,1}{4,4,1,4,3,2}
48{9,9,1}{3,1,5,4,5,5,2,4}
49{2,11,1}{2,1,1,2}
50{7,7,1}{2,3,3,2}{2,2,2,2,1,2}
51{9,3,2}{3,2,2,3,3,2}
52{1,13,1}{1,2,1,1,3,2}{2,2,2,1,1,2,2}
53{5,11,1}{3,2,3,2}
54{13,1}{2,3,2,3,2}
55{3,3,3}
56{3,18,1}{3,1,3,4,2}
57{1,37,1}{5,1,4,2}{1,4,4,2,4,2}
58{1,6,2}{1,1,2,2,2}
59{1,16,1}{1,3,5,3,2}{1,3,3,3,3,3,2}
60{9,13,1}{2,7,4,2}{3,4,4,4,2}
61{2,25,1}{3,2,2,3,2}{2,2,2,2,2,1,2,1,2,2,2}
62{1,11,1}{3,2,5,2}{4,1,4,4,2}
63{14,1}{1,2,5,3,3}{2,3,2,4,4,3}
64{3,19,1}{4,1,1,3}{2,2,2,2,1,1,2}
65{23,1}{1,1,2,4,2}{2,2,2,2,2,2,1,2,1,2,2}
66{4,2,3}{3,3,2,3,3,3}
67{14,1}{4,3,5,2}{2,3,4,3,2,4,2}
68{1,19,1}{4,1,2,2}{3,3,2,2,2}
69{5,17,1}{2,5,2,2}{3,3,2,3,2}
70{3,26,1}{4,3,2,4,2}
71{1,13,1}{1,4,1,6,2}{1,2,3,2,3,4,2}
72{13,1}{1,6,3,2}{2,2,1,3,3,2,2}
73{15,1}{2,6,1,2}{2,3,1,3,3,2,2}
74{17,1}{2,8,2}{2,3,3,2,2,3,3}
75{19,1}{2,2,2,2,5,2}{2,4,4,4,4,2}
76{2,19,1}{6,2,1,4,2}{4,4,4,2,4 2}
77{1,25,1}{2,1,1,6,2}{3,2,3 4,3,2}
78{1,25,1}{1,3,2,4,2}
79{4,21,1}{4,1,2,5,4,2}
80{17,1}{5,1,3,3}{3,3,2,3,3}
81{1,2}
82{5,1,4}{2,4,1,2}{3,3,2,2}
83{5,31,1}{4,4,2,4}
84{4,30,1}{1,2,2,5,2}{3,1,3,4,4,2}
85{3,20,1}{2,1,2,2,2,2}
86{8,2,2}{3,3,3,3,3,2,2}
87{7,21,1}{3,1,2,4,2}
88{21,1}{1,2,4,2}{1,2,2,3,3,3,3,2}
89{1,22,1}{1,2,2,2,2,2}
90{19,1}{3,4,2,2,2}
91{17,1}{2,3,1,3,2,3}
92{7,21,1}{1,1,3,3,3,3}
93{1,55,1}{2,3,4,2,2}
94{5,2}{3,1,3,2,3,2}
95{5,25,1}{1,8,5,2}{3,2,1,4,4,3,4,2}
96{7,21,1}{3,1,3,6,2}{5,4,5,2}
97{25,1}{1,1,2,2}
98{1,25,1}{2,4,2,4}{2,3,3,2,3,3,3,2}
99{21,1}{2,5,4,2}{2,3,3,3,3,3}

Luke Peabody found that 2008{2,512,1}=2008 and 2008{5,1,7,7,6,7,3}=2008. Can anyone find a shorter list or one with a small sum? He claims no list using numbers smaller than 7 will work.

2. Here are the smallest known differences for n ≤ 24:

Minimal Power Differences
2nkSmallest
Difference
Expression
414( 23 ) – ( 41 )
611( 15 + 26 ) – ( 43 )
8116( 16 + 27 + 83 ) – ( 54 )
29( 27 + 83 ) – ( 54 + 61 )
10128( 19 + 37 + 65 + 82 ) – ( 104 )
2–1( 56 + 84 + 101 ) – ( 39 + 72 )

Smallest Known Power Differences
2nkSmallest
Difference
Expression
121–157( 311 + 57 + 94 + 101 + 122 ) – ( 86 )
2–2( 211 + 39 + 105 + 121 ) – ( 76 + 84 )
3–15( 211 + 39 + 105 ) – ( 112 + 76 + 84 )
1412( 214 + 411 + 78 + 105 + 121 + 133 ) – ( 69 )
2–1( 111 + 314 + 59 + 122 + 134 ) – ( 78 + 106 )
32( 310 + 413 + 611 + 75 ) – ( 114 + 92 + 128 )
161444( 112 + 215 + 314 + 410 + 511 + 137 + 166 ) – ( 89 )
2–26( 105 + 114 + 131 + 143 + 152 + 169 ) – ( 76 + 812 )
34( 215 + 314 + 410 ) – ( 511 + 89 + 121 + 137 + 166 )
417( 112 + 215 + 314 + 410 ) – ( 511 + 89 + 137 + 166 )
1811386( 116 + 214 + 315 + 410 + 611 + 127 + 139 + 175 ) – ( 188 )
2–93( 412 + 711 + 135 + 148 + 151 + 166 + 173 ) – ( 218 + 910 )
3–109( 412 + 711 + 135 + 148 + 166 + 173 ) – ( 115 + 218 + 910 )
45( 109 + 112 + 137 + 151 + 176 ) – ( 318 + 414 + 128 + 165 )
201–8736( 314 + 812 + 116 + 151 + 169 + 177 + 182 + 195 + 204 ) – ( 1310 )
294( 119 + 318 + 515 + 1110 + 139 + 162 + 174 + 207 ) – ( 812 + 146 )
374( 318 + 515 + 1110 + 139 + 162 + 174 + 207 ) – ( 812 + 146 + 191 )
2213278( 122 + 221 + 319 + 420 + 138 + 149 + 156 + 167 + 1710 + 185 ) – ( 1112 )
23255( 221 + 319 + 420 + 138 + 149 + 156 + 167 + 1710 + 185 ) – ( 1112 + 221)
3–24575( 220 + 321 + 1211 + 151 + 169 + 188 + 196 + 225 ) – ( 417 + 714 + 1310 )
241–5359( 222 + 421 + 1311 + 1412 + 156 + 167 + 178 + 185 + 191 + 209 + 233 ) – ( 2410 )
2–5379( 222 + 421 + 1311 + 1412 + 156 + 167 + 178 + 185 + 209 + 233 ) – ( 119 + 2410 )

3. Here are the smallest known results:

Smallest Collection of k Different nth Powers with the Same Digits
k \ n234
2 132 = 169
142 = 196
53 = 125
83 = 512
44 = 256
54 = 625
3 132 = 169
142 = 196
312 = 961
3453 = 41063625
3843 = 56623104
4053 = 66430125
10014 = 1004006004001
10104 = 1040604010000
11004 = 1464100000000
4 1282 = 16384
1782 = 31684
1912 = 36481
1962 = 38416
10023 = 1006012008
10203 = 1061208000
20013 = 8012006001
20103 = 8120601000
100014 = 10004000600040001
100104 = 10040060040010000
101004 = 10406040100000000
110004 = 14641000000000000
5 1282 = 16384
1782 = 31684
1912 = 36481
1962 = 38416
2092 = 43681
50273 = 127035954683
70613 = 352045367981
72023 = 373559126408
82883 = 569310543872
83843 = 589323567104
186374 = 120643525773897361
228764 = 273854796251013376
247884 = 377542589207163136
270114 = 532307581397762641
275064 = 572413350873761296
6 1032 = 10609
1302 = 16900
1402 = 19600
2472 = 61009
3012 = 90601
3102 = 96100
112573 = 1426487591593
112723 = 1432197595648
151783 = 3496581419752
162313 = 4275981654391
168853 = 4813967954125
196543 = 7591941538264
7 12802 = 1638400
17802 = 3168400
19102 = 3648100
19512 = 3806401
19602 = 3841600
20092 = 4036081
20902 = 4368100
233953 = 12804692354875
235723 = 13097526845248
249283 = 15490388426752
254293 = 16443257028589
258163 = 17205482538496
337393 = 38405782562419
349273 = 42607284155983
8 10032 = 1006009
10302 = 1060900
13002 = 1690000
14002 = 1960000
24702 = 6100900
30012 = 9006001
30102 = 9060100
31002 = 9610000

k \ n56
2 3485 = 5103830227968
3815 = 8028323765901
7316 = 152582336769287881
7646 = 198865822812737536
3 100015 = 100050010001000050001
100105 = 100501001000500100000
101005 = 105101005010000000000
100016 = 1000600150020001500060001
100106 = 1006015020015006001000000
101006 = 1061520150601000000000000
4 775775 = 2809732043385544072671657
822065 = 3754201540342892587063776
847445 = 4370637587630405589172224
883265 = 5375796440820402273185376
1000016 = 1000060001500020000150000600001
1000106 = 1000600150020001500060001000000
1001006 = 1006015020015006001000000000000
1010006 = 1061520150601000000000000000000
5 1143185 = 19524192871006530732473568
1154315 = 20493318522670793075846151
1164185 = 21384577307426399005121568
1331225 = 41807015820653779495321632
1412135 = 56153056098471782041237293

4. Here are the known exponent switch equations with 5 or fewer terms on each side:

Jeremy Galvagni noted that these are too easy if we allow 1, or repeats of terms.

Known Small Exponent Switch Equations
 24 = 42 25 + 27 + 29 + 53 + 54 = 52 + 72 + 92 + 35 + 45 25 + 26 + 27 + 45 + 63 = 52 + 62 + 72 + 54 + 36 23 + 27 + 36 + 54 + 82 = 32 + 72 + 63 + 45 + 28 25 + 26 + 211 + 53 + 73 = 52 + 62 + 112 + 35 + 37 23 + 29 + 211 + 63 + 73 = 32 + 92 + 112 + 36 + 37 45 + 46 + 73 + 92 + 102 = 54 + 64 + 37 + 29 + 210 26 + 210 + 211 + 64 + 72 = 62 + 102 + 112 + 46 + 27 23 + 28 + 43 + 56 + 132 = 32 + 82 + 34 + 65 + 213 28 + 310 + 45 + 46 + 84 = 82 + 103 + 54 + 64 + 48 212 + 216 + 46 + 74 + 103 = 122 + 162 + 64 + 47 + 310 25 + 216 + 43 + 75 + 122 = 52 + 162 + 34 + 57 + 212 29 + 212 + 220 + 74 + 104 = 92 + 122 + 202 + 47 + 410

Dean Hickerson asked what is the smallest K so that there are infinitely many exponent switch equations with K or fewer terms on each side? And then he noted that K ≤ 20 since:

22n+ 22n+8+ 22n+16+ 22n+32+ 22n+34 + 4n+1+ 4n+2+ 4n+10+ 4n+14+ 4n+18 +
n4 + (n+4)4 + (n+8)4 + (n+16)4 + (n+17)4 + (2n+2)2 + (2n+4)2 + (2n+20)2 + (2n+28)2 + (2n+36)2 =
(2n)2 + (2n+8)2 + (2n+16)2 + (2n+32)2 + (2n+34)2 + (n+1)4 + (n+2)4 + (n+10)4 + (n+14)4 + (n+18)4 +
4n + 4n+4+ 4n+8+ 4n+16+ 4n+17 + 22n+2+ 22n+4+ 22n+20+ 22n+28+ 22n+36.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/29/08.