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 k^th 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, ... n_k}} = 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 ± a_i^b_i be to 0 if the a_i and b_i 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 n^th 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 x^y = y^x. 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?

ANSWERS

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

Smallest GRDI Exponents

n	min length	min sum	min max
10	none
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 Pebody 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

2n	k	Smallest Difference	Expression
4	1	4	( 2³ ) – ( 4¹ )
6	1	1	( 1⁵ + 2⁶ ) – ( 4³ )
8	1	16	( 1⁶ + 2⁷ + 8³ ) – ( 5⁴ )
8	2	9	( 2⁷ + 8³ ) – ( 5⁴ + 6¹ )
10	1	28	( 1⁹ + 3⁷ + 6⁵ + 8² ) – ( 10⁴ )
10	2	–1	( 5⁶ + 8⁴ + 10¹ ) – ( 3⁹ + 7² )

Smallest Known Power Differences

2n	k	Smallest Difference	Expression
12	1	–157	( 3¹¹ + 5⁷ + 9⁴ + 10¹ + 12² ) – ( 8⁶ )
	2	–2	( 2¹¹ + 3⁹ + 10⁵ + 12¹ ) – ( 7⁶ + 8⁴ )
	3	–15	( 2¹¹ + 3⁹ + 10⁵ ) – ( 1¹² + 7⁶ + 8⁴ )
14	1	2	( 2¹⁴ + 4¹¹ + 7⁸ + 10⁵ + 12¹ + 13³ ) – ( 6⁹ )
	2	–1	( 1¹¹ + 3¹⁴ + 5⁹ + 12² + 13⁴ ) – ( 7⁸ + 10⁶ )
	3	2	( 3¹⁰ + 4¹³ + 6¹¹ + 7⁵ ) – ( 1¹⁴ + 9² + 12⁸ )
16	1	444	( 1¹² + 2¹⁵ + 3¹⁴ + 4¹⁰ + 5¹¹ + 13⁷ + 16⁶ ) – ( 8⁹ )
	2	–26	( 10⁵ + 11⁴ + 13¹ + 14³ + 15² + 16⁹ ) – ( 7⁶ + 8¹² )
	3	4	( 2¹⁵ + 3¹⁴ + 4¹⁰ ) – ( 5¹¹ + 8⁹ + 12¹ + 13⁷ + 16⁶ )
	4	17	( 1¹² + 2¹⁵ + 3¹⁴ + 4¹⁰ ) – ( 5¹¹ + 8⁹ + 13⁷ + 16⁶ )
18	1	1386	( 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 )
	4	5	( 10⁹ + 11² + 13⁷ + 15¹ + 17⁶ ) – ( 3¹⁸ + 4¹⁴ + 12⁸ + 16⁵ )
20	1	–8736	( 3¹⁴ + 8¹² + 11⁶ + 15¹ + 16⁹ + 17⁷ + 18² + 19⁵ + 20⁴ ) – ( 13¹⁰ )
	2	94	( 1¹⁹ + 3¹⁸ + 5¹⁵ + 11¹⁰ + 13⁹ + 16² + 17⁴ + 20⁷ ) – ( 8¹² + 14⁶ )
	3	74	( 3¹⁸ + 5¹⁵ + 11¹⁰ + 13⁹ + 16² + 17⁴ + 20⁷ ) – ( 8¹² + 14⁶ + 19¹ )
22	1	3278	( 1²² + 2²¹ + 3¹⁹ + 4²⁰ + 13⁸ + 14⁹ + 15⁶ + 16⁷ + 17¹⁰ + 18⁵ ) – ( 11¹² )
	2	3255	( 2²¹ + 3¹⁹ + 4²⁰ + 13⁸ + 14⁹ + 15⁶ + 16⁷ + 17¹⁰ + 18⁵ ) – ( 11¹² + 22¹)
	3	–24575	( 2²⁰ + 3²¹ + 12¹¹ + 15¹ + 16⁹ + 18⁸ + 19⁶ + 22⁵ ) – ( 4¹⁷ + 7¹⁴ + 13¹⁰ )
24	1	–5359	( 2²² + 4²¹ + 13¹¹ + 14¹² + 15⁶ + 16⁷ + 17⁸ + 18⁵ + 19¹ + 20⁹ + 23³ ) – ( 24¹⁰ )
24	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 n^th Powers with the Same Digits

k \ n	2	3	4
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 \ n 5 6
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

k \ n	5	6
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

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

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

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.