Problem of the Month
(April 2009)

This month's problem comes from Kevin Savage, who noticed that 72 × 54 × 36 × 9 × 8 × 1 = 69. He asked what other powers could be written as a product using one each of the digits 1-9. The same question could also be asked about the digits 0-9.

Suppose no power of n can be written as a product using one of each digit. What is the smallest k for which some power of n can be factored using exactly k of each digit? For example, 6751269 × 823543 × 81 × 49 × 7 = 2113, using two of each digit 1-9.

Another way to generalize this is to ask which numbers have two different factorizations, one using each digit once, and one using only copies of one digit. This would include solutions such as 5476 × 198 × 32 = 2 × 2 × 2 × 2 × 2 × 22 × 222 × 222.

What are the answers in other bases?

ANSWERS

Here are the powers with bases smaller than 100 and the factorizations using k of each digit for the smallest known k. (The zeroes in green can be placed in many other permutations.)

Powers with Equidigital Factorizations
nUsing 1-9Using 0-9
2 see below 536870912 × 4 = 231
3 see below see below
4 see below 536870912 × 536870912 × 4 × 4 = 431
5 ≥17 ≥10
6 72 × 54 × 36 × 9 × 8 × 1 = 69
(Kevin Savage)
576 × 81 × 9 × 4 × 3 × 2 = 69
576 × 243 × 9 × 8 × 1 = 69
3456 × 972 × 18 = 610
5184 × 72 × 9 × 6 × 3 = 610 		3456 × 978 × 108 = 611
10368 × 72 × 54 × 9 = 611
7 ≥17 ≥10
8 see below ≥ 2
9 ≥5 see below
10 ≥3≥3
11 ≥17 ≥10
12 576 × 24 × 9 × 8 × 3 × 1 = 126
1536 × 729 × 8 × 4 = 127 		3072 × 1458 × 96 = 128
5308416 × 972 = 129
13 ≥17 ≥10
14 537824 × 196 = 147 470596 × 19208 × 343 × 128 × 56 × 7 = 1415
470596 × 25088 × 343 × 196 × 7 × 2 × 1 = 1414
470596 × 307328 × 56 × 49 × 28 × 1 × 1 = 1414
7529536 × 43904 × 16807 × 28 × 1 = 1415
8605184 × 470596 × 1372 × 392 = 1416
15059072 × 686 × 343 × 128 × 49 × 7 = 1415
15059072 × 76832 × 38416 × 49 = 1416
40353607 × 1792 × 98 × 56 × 28 × 14 = 1415
40353607 × 1792 × 1568 × 49 × 28 = 1415
40353607 × 5488 × 1792 × 196 × 2 = 1415
564950498 × 17210368 × 32 × 7 = 1416
253097823104 × 784 × 196 × 56 = 1416
775112083256 × 43904 × 896 = 1417
15 ≥3≥3
16 see below ≥ 2
17 ≥17 ≥10
18 54 × 27 × 9 × 8 × 6 × 3 × 1 = 185
96 × 81 × 54 × 27 × 3 = 186
972 × 54 × 36 × 18 = 186
1458 × 72 × 36 × 9 = 186
2187 × 96 × 54 × 3 = 186 		104976 × 5832 = 187
19 ≥17 ≥10
20 ≥3≥3
21 6751269 × 823543 × 81 × 49 × 7 = 2113 453789 × 64827 × 21609 × 5103 = 2114
68641485507 × 1029 × 27 × 9 × 3 × 3 = 2113
22 2357947691 × 5324 × 16 × 8 × 8 = 2212 ≥ 3
23 ≥17 ≥10
24 576 × 48 × 32 × 9 × 1 = 245
576 × 128 × 9 × 4 × 3 = 245 		3072 × 54 × 16 × 9 × 8 = 246
25 ≥17 ≥10
26 ≥ 3 9653618 × 70304 × 2197 × 52 × 8 × 4 = 2613
27 see below see below
28 98 × 56 × 32 × 14 × 7 = 285
(Richard Sabey)
392 × 56 × 14 × 8 × 7 = 285
(Richard Sabey)
784 × 392 × 56 × 1 = 285
(Richard Sabey)
1568 × 49 × 32 × 7 = 285
(Richard Sabey)
1568 × 392 × 7 × 4 = 285
(Richard Sabey) 		25088 × 10976 × 343 × 256 × 49 × 7 × 1 = 2811
50176 × 9604 × 5488 × 392 × 32 × 7 × 1 = 2812
50176 × 19208 × 343 × 256 × 49 × 8 × 7 = 2812
50176 × 19208 × 343 × 256 × 98 × 7 × 4 = 2812
50176 × 50176 × 343 × 98 × 49 × 28 × 2 = 2812
100352 × 38416 × 256 × 98 × 49 × 7 × 7 = 2812
307328 × 9604 × 512 × 56 × 49 × 8 × 7 × 1 = 2812
307328 × 9604 × 512 × 98 × 56 × 7 × 4 × 1 = 2812
307328 × 12544 × 10976 × 98 × 56 = 2812
823543 × 50176 × 4096 × 98 × 7 × 2 × 1 = 2812
2809856 × 9604 × 512 × 343 × 7 × 7 × 1 = 2812
15059072 × 3136 × 64 × 49 × 28 × 8 × 7 = 2812
15059072 × 3136 × 98 × 64 × 28 × 7 × 4 = 2812
40353607 × 8192 × 256 × 49 × 8 × 7 × 1 = 2812
40353607 × 8192 × 256 × 98 × 7 × 4 × 1 = 2812
29 ≥17 ≥10
30 9375 × 24 × 18 × 6 = 305
84375 × 16 × 9 × 2 = 305 		18750 × 9 × 6 × 4 × 3 × 2 = 305
93750 × 162 × 48 = 306
93750 × 648 × 12 = 306
31 ≥17 ≥10
32 see below ≥ 2
33 ≥3≥3
34 ≥3≥3
35 64339296875 × 42875 × 1 × 1 = 3510 ≥ 3
36 3456 × 972 × 18 = 365
5184 × 72 × 9 × 6 × 3 = 365 		≥ 2
37 ≥17 ≥10
38 39617584 × 2 = 385 39617584 × 39617584 × 2 × 2 = 3810
39 ≥ 3 4826809 × 4563 × 507 × 27 × 9 × 3 × 1 × 1 = 3910
40 ≥3≥3
41 ≥17 ≥10
42 756 × 49 × 28 × 3 × 1 = 424 504 × 98 × 21 × 7 × 6 × 3 = 425
504 × 126 × 98 × 7 × 3 = 425
3087 × 56 × 21 × 9 × 4 = 425
3087 × 98 × 56 × 7 × 1 = 425
19208 × 54 × 7 × 6 × 3 = 425
19208 × 567 × 4 × 3 = 425
43 ≥17 ≥10
44 77948684 × 1936 × 512 × 352 = 4410 937024 × 58564 × 30976 × 8 × 2 × 1 × 1 = 4410
45 4782969 × 84375 × 625 × 3 × 1 × 1 = 459 87890625 × 43046721 × 15 × 9 × 3 = 4511
474609375 × 19683 × 2025 × 81 = 4511
46 ≥3≥3
47 ≥17 ≥10
48 576 × 32 × 9 × 8 × 4 × 1 = 484
24576 × 9 × 8 × 3 × 1 = 484 		≥ 2
49 ≥17 ≥10
50 ≥3≥3
51 ≥3≥3
52 ≥ 3 5429503678976 × 104 × 32 × 8 × 1 = 5210
5429503678976 × 2048 × 13 × 1 = 5210
53 ≥17 ≥10
54 54 × 36 × 27 × 18 × 9 = 544 ≥ 2
55 ≥3≥3
56 3584 × 196 × 7 × 2 = 564 19208 × 4096 × 412 × 343 × 56 × 8 × 7 × 7 = 5610
50176 × 1024 × 343 × 256 × 98 × 98 × 7 = 5610
470596 × 25088 × 14336 × 1792 = 5610
470596 × 351232 × 4096 × 8 × 8 × 7 × 1 = 5610
15059072 × 14336 × 896 × 784 × 2 = 5610
58720256 × 10976 × 343 × 98 × 14 = 5610
30840979456 × 1372 × 128 × 56 = 5610
57 ≥3≥3
58 ≥3≥3
59 ≥17 ≥10
60 1875 × 32 × 9 × 6 × 4 = 604
1875 × 96 × 24 × 3 = 604
84375 × 9216 = 605 		3750 × 9 × 8 × 6 × 4 × 2 × 1 = 604
18750 × 432 × 96 = 605
61 ≥17 ≥10
62 ≥3≥3
63 583443 × 9261 × 567 × 189 × 27 = 639 64827 × 15309 = 635
64 see below ≥ 2
65 ≥3≥3
66 ≥ 2 790614 × 528 × 3 = 665
69574032 × 18 = 665
67 ≥17 ≥10
68 ≥ 3 96550276 × 17408 × 4913 × 32 × 8 = 6810
69 839523 × 128547 × 4761 × 69 = 699 328509 × 4761 = 695
70 4375 × 196 × 28 = 704 9604 × 3125 × 8 × 7 = 705
71 ≥17 ≥10
72 1728 × 96 × 54 × 3 = 724
3456 × 972 × 8 × 1 = 724 		≥ 2
73 ≥17 ≥10
74 ≥ 3 239892608 × 50653 × 74 × 74 × 1 × 1 = 749
75 ≥3≥3
76 877952 × 6859 × 361 × 32 × 4 × 4 × 1 = 768 23104 × 6859 × 5776 × 304 × 19 × 8 × 2 = 769
77 ≥3≥3
78 6591 × 78 × 24 × 3 = 784
(Bryce Herdt)
39546 × 78 × 12 = 784 		507 × 39 × 26 × 18 × 4 = 784
257049 × 8 × 6 × 3 × 1 = 784
79 ≥17 ≥10
80 ≥3≥3
81 ≥5 see below
82 ≥3≥3
83 ≥17 ≥10
84 81 × 56 × 49 × 32 × 7 = 784
392 × 81 × 56 × 7 × 4 = 784
3456 × 98 × 21 × 7 = 784
3528 × 147 × 96 = 784 		7056 × 98 × 24 × 3 × 1 = 784
7056 × 294 × 8 × 3 × 1 = 784
10976 × 54 × 28 × 3 = 784
85 ≥3≥3
86 ≥ 3 159014 × 79507 × 86 × 86 × 43 × 32 × 2 = 869
87 ≥3≥3
88 5153632 × 8192 × 7744 × 968 = 889 10903552 × 3872 × 968 × 176 × 44 = 889
89 ≥17 ≥10
90 16875 × 432 × 9 = 904 ≥ 2
91 ≥3≥3
92 736 × 529 × 184 = 924 ≥ 2
93 ≥3≥3
94 ≥3≥3
95 ≥3≥3
96 4718592 × 6 × 3 = 964 ≥ 2
97 ≥17 ≥10
98 ≥ 3 823543 × 10976 × 2401 × 98 × 56 × 7 = 989
823543 × 470596 × 19208 × 16 × 7 = 989
11529602 × 470596 × 343 × 8 × 8 × 7 = 989
40353607 × 98 × 98 × 56 × 14 × 7 × 2 × 2 × 1 = 988
40353607 × 1568 × 98 × 49 × 7 × 2 × 2 × 1 = 988
99 ≥3≥3

Here are the solutions for k≥4 that wouldn't fit nicely in the above table:

Prime Powers with Equidigital Factorizations Using 1-9
nMin kExpression
2, 4, 8, 16, 32, 645 17179869184 × 8589934592 × 33554432 × 16777216 × 256 × 64 × 32 × 8 = 2138 = 469 = 846 = 6423
137438953472 × 8589934592 × 16777216 × 65536 × 8192 × 16 × 8 × 4 × 4 × 2 = 2135 = 845 = 3227
137438953472 × 8589934592 × 16777216 × 65536 × 8192 × 64 × 8 × 4 × 2 × 1 = 2135 = 845 = 3227
137438953472 × 8589934592 × 16777216 × 65536 × 8192 × 128 × 64 × 4 = 2138 = 469 = 846 = 6423
137438953472 × 8589934592 × 268435456 × 16777216 × 8192 × 16 = 2139
137438953472 × 68719476736 × 8589934592 × 256 × 256 × 8 × 4 × 2 × 1 × 1 × 1 = 2128 = 464 = 1632
137438953472 × 68719476736 × 8589934592 × 256 × 256 × 128 × 4 × 1 × 1 = 2131
137438953472 × 68719476736 × 8589934592 × 512 × 256 × 16 × 8 × 4 × 2 × 1 = 2133
137438953472 × 68719476736 × 8589934592 × 512 × 256 × 64 × 8 × 2 × 1 × 1 = 2133
137438953472 × 68719476736 × 8589934592 × 512 × 256 × 128 × 16 × 4 = 2136 = 468 = 1634
137438953472 × 68719476736 × 8589934592 × 512 × 256 × 128 × 64 × 1 = 2136 = 468 = 1634
137438953472 × 68719476736 × 8589934592 × 512 × 512 × 64 × 16 × 8 × 2 = 2138 = 469 = 846 = 6423
3, 274 7625597484987 × 1594323 × 19683 × 6561 × 243 × 81 × 27 = 369 = 2723
7625597484987 × 1594323 × 19683 × 6561 × 2187 × 243 = 369 = 2723

Prime Powers with Equidigital Factorizations Using 0-9
nMin kExpression
3, 9, 27, 816 4052555153018976267 × 10460353203 × 3486784401 × 4782969 × 19683 × 81 × 27 × 27 × 9 × 9 = 3117 = 2739
4052555153018976267 × 10460353203 × 3486784401 × 4782969 × 19683 × 729 × 81 × 27 × 9 = 3118 = 959
4052555153018976267 × 10460353203 × 3486784401 × 4782969 × 19683 × 729 × 729 × 81 = 3119
4052555153018976267 × 10460353203 × 3486784401 × 4782969 × 19683 × 2187 × 27 × 9 × 9 = 3117 = 2739
4052555153018976267 × 10460353203 × 3486784401 × 4782969 × 19683 × 2187 × 729 × 9 = 3118 = 959
4052555153018976267 × 847288609443 × 31381059609 × 43046721 × 19683 × 27 × 27 × 9 = 3119
4052555153018976267 × 847288609443 × 31381059609 × 43046721 × 19683 × 729 × 27 = 3120 = 960 = 2740 = 8130
26588814358957503287787 × 10460353203 × 1162261467 × 43046721 × 59049 × 9 × 9 × 9 = 3119
92709463147897837085761925410587 × 282429536481 × 10460353203 × 6561 × 9 = 3122 = 961

Here are the ways to factor a number using one of each digit or only one type of digit. (The digits in red can be omitted to give factorizations of all but one digit equal to factorizations of only that digit.)

Equidigital Factorizations of Monodigital Factorizations
nUsing 1-9Using 0-9
2 256 × 148 × 37 × 9 = 28 × 2222
352 × 96 × 74 × 8 × 1 = 212 × 22 × 222
512 × 96 × 37 × 8 × 4 = 218 × 222
592 × 48 × 37 × 6 × 1 = 27 × 2222
592 × 64 × 37 × 18 = 29 × 2222
592 × 74 × 8 × 6 × 3 × 1 = 27 × 2222
592 × 74 × 36 × 8 × 1 = 28 × 2222
592 × 176 × 8 × 4 × 3 = 211 × 22 × 222
592 × 384 × 176 = 213 × 22 × 222
814 × 256 × 37 × 9 = 26 × 22 × 2222
968 × 512 × 74 × 3 = 210 × 222 × 222
1452 × 968 × 37 = 224 × 222
1584 × 296 × 37 = 24 × 22 × 2222
1936 × 528 × 74 = 25 × 223 × 222
1968 × 542 × 37 = 23 × 222 × 22222
3168 × 592 × 74 = 27 × 22 × 2222
3256 × 74 × 9 × 8 × 1 = 24 × 22 × 2222
3256 × 198 × 74 = 2 × 222 × 2222
4736 × 592 × 18 = 210 × 2222
5476 × 32 × 9 × 8 × 1 = 28 × 2222
5476 × 192 × 8 × 3 = 29 × 2222
5476 × 198 × 32 = 25 × 22 × 2222
6512 × 37 × 9 × 8 × 4 = 26 × 22 × 2222
8954 × 37 × 12 × 6 = 222 × 2222
8954 × 176 × 3 × 2 = 22 × 223 × 222
9472 × 1536 × 8 = 219 × 222
9768 × 352 × 4 × 1 = 27 × 222 × 222
9768 × 5324 × 1 = 224 × 222
32856 × 74 × 9 × 1 = 2 × 2223
35816 × 74 × 9 × 2 = 2 × 222 × 2222
53724 × 968 × 1 = 224 × 222
61952 × 48 × 37 = 210 × 222 × 222
61952 × 74 × 8 × 3 = 210 × 222 × 222 		407 × 352 × 96 × 8 × 1 = 210 × 222 × 222
592 × 407 × 8 × 6 × 3 × 1 = 25 × 22 × 2222
592 × 407 × 36 × 8 × 1 = 26 × 22 × 2222
704 × 592 × 16 × 8 × 3 = 215 × 22 × 222
968 × 512 × 407 × 3 = 28 × 223 × 222
968 × 704 × 352 × 1 = 210 × 224
1936 × 528 × 407 = 23 × 224 × 222
3168 × 592 × 407 = 25 × 222 × 2222
3256 × 407 × 9 × 8 × 1 = 22 × 222 × 2222
4096 × 528 × 37 × 1 = 214 × 22 × 222
4107 × 352 × 96 × 8 = 210 × 22 × 2222
4107 × 592 × 8 × 6 × 3 = 25 × 2223
4107 × 592 × 36 × 8 = 26 × 2223
4107 × 3256 × 9 × 8 = 22 × 22 × 2223
4608 × 592 × 37 × 1 = 211 × 2222
5476 × 1089 × 32 = 23 × 222 × 2222
7104 × 968 × 352 = 210 × 223 × 222
8954 × 3072 × 16 = 212 × 222 × 222
10952 × 768 × 4 × 3 = 211 × 2222
10952 × 864 × 37 = 25 × 2223
17908 × 256 × 4 × 3 = 29 × 222 × 222
17908 × 352 × 6 × 4 = 26 × 223 × 222
19536 × 704 × 8 × 2 = 211 × 222 × 222
30976 × 512 × 8 × 4 = 220 × 222
52096 × 37 × 18 × 4 = 27 × 22 × 2222
61952 × 407 × 8 × 3 = 28 × 223 × 222
90354 × 726 × 8 × 1 = 223 × 2222
90354 × 768 × 2 × 1 = 27 × 22 × 2222
98304 × 65712 = 217 × 2222
2150896 × 74 × 3 = 2 × 222 × 222 × 2222
3748096 × 512 = 213 × 224
536870912 × 4 = 231
4380756192 = 22 × 2222 × 22222
3   24057 × 19683 = 315 × 33
406593 × 81 × 27 = 35 × 33 × 3332
406593 × 2187 = 35 × 33 × 3332
41065893 × 27 = 3 × 3332 × 3333
4 256 × 148 × 37 × 9 = 43 × 4442
352 × 96 × 74 × 8 × 1 = 45 × 44 × 444
592 × 74 × 36 × 8 × 1 = 43 × 4442
3168 × 592 × 74 = 42 × 44 × 4442
4736 × 592 × 18 = 44 × 4442
5476 × 32 × 9 × 8 × 1 = 43 × 4442
5476 × 198 × 32 = 4 × 44 × 4442
9472 × 1536 × 8 = 49 × 444
9768 × 352 × 4 × 1 = 42 × 442 × 444 		592 × 407 × 8 × 6 × 3 × 1 = 4 × 44 × 4442
968 × 512 × 407 × 3 = 42 × 443 × 444
968 × 704 × 352 × 1 = 43 × 444
4096 × 528 × 37 × 1 = 46 × 44 × 444
4107 × 592 × 8 × 6 × 3 = 4 × 4443
7104 × 968 × 352 = 43 × 443 × 444
10952 × 864 × 37 = 4 × 4443
17908 × 256 × 4 × 3 = 43 × 442 × 444
17908 × 352 × 6 × 4 = 4 × 443 × 444
19536 × 704 × 8 × 2 = 44 × 442 × 444
30976 × 512 × 8 × 4 = 49 × 442
52096 × 37 × 18 × 4 = 42 × 44 × 4442
61952 × 407 × 8 × 3 = 42 × 443 × 444
90354 × 768 × 2 × 1 = 42 × 44 × 4442
5   840269375 × 1 = 5 × 552 × 55555
6 72 × 54 × 36 × 9 × 8 × 1 = 69
96 × 81 × 54 × 37 × 2 = 66 × 666
216 × 54 × 37 × 9 × 8 = 66 × 666
576 × 81 × 9 × 4 × 3 × 2 = 69
576 × 243 × 9 × 8 × 1 = 69
726 × 594 × 8 × 3 × 1 = 62 × 663
1369 × 54 × 27 × 8 = 62 × 6662
1458 × 296 × 37 = 62 × 6662
1584 × 72 × 9 × 6 × 3 = 67 × 66
1628 × 54 × 37 × 9 = 66 × 6662
1728 × 594 × 6 × 3 = 67 × 66
2178 × 96 × 54 × 3 = 65 × 662
2376 × 1584 × 9 = 65 × 662
3168 × 72 × 54 × 9 = 68 × 66
3256 × 81 × 74 × 9 = 6 × 66 × 6662
3456 × 198 × 27 = 67 × 66
3456 × 297 × 18 = 67 × 66
3456 × 972 × 18 = 610
3564 × 72 × 9 × 8 × 1 = 67 × 66
4356 × 198 × 72 = 63 × 663
4356 × 297 × 8 × 1 = 62 × 663
4356 × 792 × 18 = 63 × 663
4356 × 972 × 8 × 1 = 65 × 662
4752 × 96 × 81 × 3 = 68 × 66
4752 × 198 × 36 = 65 × 662
4752 × 396 × 18 = 65 × 662
5184 × 72 × 9 × 6 × 3 = 610
5184 × 2376 × 9 = 68 × 66
5346 × 792 × 8 × 1 = 65 × 662
5476 × 891 × 3 × 2 = 66 × 6662
6534 × 72 × 9 × 8 × 1 = 65 × 662
7128 × 96 × 54 × 3 = 68 × 66
7326 × 1584 × 9 = 62 × 662 × 666
7326 × 5184 × 9 = 65 × 66 × 666
21384 × 576 × 9 = 68 × 66
35816 × 729 × 4 = 62 × 662 × 666
43956 × 27 × 8 × 1 = 63 × 66 × 666
43956 × 72 × 18 = 64 × 66 × 666
47952 × 36 × 18 = 66 × 666
53946 × 72 × 8 × 1 = 66 × 666
175824 × 36 × 9 = 64 × 66 × 666
192456 × 37 × 8 = 64 × 66 × 666
351648 × 297 = 62 × 662 × 666
351648 × 972 = 65 × 66 × 666
431568 × 792 = 65 × 66 × 666
1479852 × 6 × 3 = 6 × 666 × 6666
1539648 × 72 = 68 × 66
3519648 × 27 = 63 × 66 × 6666
4319568 × 72 = 66 × 6666
6479352 × 8 × 1 = 65 × 6666
372594816 = 64 × 663 		407 × 352 × 81 × 9 × 6 = 63 × 662 × 666
1056 × 729 × 48 × 3 = 68 × 66
1089 × 576 × 324 = 66 × 662
3456 × 297 × 108 = 68 × 66
3456 × 972 × 108 = 611
4059 × 271 × 8 × 6 × 3 = 62 × 66 × 66666
4356 × 792 × 108 = 64 × 663
4752 × 396 × 108 = 66 × 662
5832 × 407 × 16 × 9 = 65 × 66 × 666
6504 × 738 × 9 × 2 × 1 = 64 × 66666
6504 × 738 × 12 × 9 = 65 × 66666
8019 × 576 × 4 × 3 × 2 = 68 × 66
8019 × 3256 × 74 = 662 × 6662
9504 × 81 × 37 × 6 × 2 = 65 × 66 × 666
9504 × 726 × 18 × 3 = 64 × 663
9504 × 2178 × 6 × 3 = 64 × 663
9504 × 2673 × 8 × 1 = 66 × 662
9504 × 7128 × 6 × 3 = 67 × 662
10368 × 72 × 54 × 9 = 611
17908 × 5346 × 2 = 663 × 666
24057 × 1936 × 8 = 64 × 663
39072 × 1458 × 6 = 65 × 66 × 666
43956 × 108 × 72 = 65 × 66 × 666
47952 × 108 × 36 = 67 × 666
47952 × 13068 = 63 × 662 × 666
57024 × 198 × 6 × 3 = 66 × 662
76032 × 54 × 18 × 9 = 69 × 66
128304 × 576 × 9 = 69 × 66
295704 × 18 × 6 × 3 = 63 × 6662
591408 × 36 × 27 = 64 × 6662
591408 × 3267 = 662 × 6662
593406 × 72 × 8 × 1 = 65 × 66 × 666
790614 × 528 × 3 = 665
1942056 × 37 × 8 = 64 × 6662
4105728 × 9 × 6 × 3 = 69 × 66
31049568 × 72 = 65 × 663
41065893 × 72 = 6662 × 6666
58074192 × 6 × 3 = 62 × 662 × 6666
69574032 × 18 = 665
3421097856 = 65 × 66 × 6666
3759820416 = 64 × 662 × 666
7 86247 × 539 × 1 = 77 × 7772 20867 × 539 × 41 = 772 × 77777
5021863 × 49 × 7 = 72 × 774
8 592 × 48 × 37 × 6 × 1 = 8 × 8882
592 × 74 × 8 × 6 × 3 × 1 = 8 × 8882
592 × 384 × 176 = 83 × 88 × 888
814 × 256 × 37 × 9 = 88 × 8882
4736 × 592 × 18 = 82 × 8882
6512 × 37 × 9 × 8 × 4 = 88 × 8882 		592 × 407 × 36 × 8 × 1 = 88 × 8882
968 × 512 × 407 × 3 = 883 × 888
4107 × 592 × 36 × 8 = 8883
8954 × 3072 × 16 = 88 × 82 × 882 × 888
17908 × 256 × 4 × 3 = 8 × 882 × 888
61952 × 407 × 8 × 3 = 883 × 888
9   24057 × 19683 = 97 × 99
406593 × 81 × 27 = 9 × 99 × 9992
406593 × 2187 = 9 × 99 × 9992

Here are the smallest solutions in some other small bases:

Powers with Equidigital Factorizations in Other Bases
nbase 2base 3base 4base 5base 6
2 10 = 101 200222 × 100111 = 2122
102020102 × 2 × 1 × 1 = 2112
102020102 × 11 × 2 = 2121
102020102 × 121 = 2122 		≥10 1003 × 13 × 4 × 4 × 2 × 2 = 231
1003 × 31 × 4 × 4 × 2 × 2 = 232
1003 × 224 × 13 × 4 = 233
1003 × 224 × 31 × 4 = 234 		101532 × 30544 × 2 = 242
34250304 × 52 × 1 × 1 = 241
3 1001 = 1110 ≥10 202023 × 1101 × 3 × 3 = 331
10303203 × 21 × 21 = 331
2001233301 × 21 = 332 		21132042 × 10404 × 3 × 3 = 334
2211023334 × 10404 = 340 		24115052350444043 × 50213 × 50213 × 213 = 3115
504423025235552210014043 × 43 × 3 × 1 × 1 × 1 = 3111
504423025235552210014043 × 43 × 13 × 1 × 1 = 3112
4 100 × 1 = 1001 102020102 × 2 × 1 × 1 = 421
102020102 × 11 × 2 = 422 		≥10 1003 × 13 × 4 × 4 × 2 × 2 = 413
1003 × 224 × 13 × 4 = 414 		101532 × 30544 × 2 = 421
5 1001110001 = 101100 210102201 = 1220 23012202233002323011 × 300311 × 1 × 1 = 11112
23012202233002323011 × 300311 × 11 = 11113 		≥10 321043523205 × 1401405 × 325 × 41 = 541
104322123314405005 × 325 × 41 × 5 = 541
6 100100 × 1 × 1 = 11010
11011000 = 11011 		20 × 1 = 201 3120 = 123 204 × 102 × 31 × 4 × 3 = 1112
1034 × 204 × 3 × 2 × 1 = 1111
10043 × 413 × 2 × 2 = 1112
40332 × 102 × 4 × 1 = 1112 		1043 × 52 = 105
7 110001 = 11110 201200112020012021 × 21 × 1 = 21102 1122033002033203213 × 211201 × 301 = 13103
1122033002033203213 × 10012213 × 1 = 13102
310321032003021212322013 × 301 × 1 = 13103 		≥10 ≥10
8 1000 × 1 × 1 = 10001 102020102 × 1012 × 1012 × 2 = 2222
102020102 × 100111 × 2 × 2 × 2 = 2222 		≥10 1003 × 224 × 13 × 4 = 1311 203504 × 30544 × 1104 × 332 × 52 × 52 × 1 × 1 = 1225
34250304 × 101532 × 30544 × 52 × 5 × 1 × 1 = 1225
1550104015504 × 332 × 332 × 12 × 4 × 4 × 2 = 1230
1550104015504 × 332 × 332 × 24 × 24 × 1 = 1230
1550104015504 × 332 × 332 × 144 × 2 × 2 = 1230
9 1001 = 10011 ≥10 2001233301 × 21 = 2113 2211023334 × 10404 = 1420 504423025235552210014043 × 43 × 13 × 1 × 1 = 1334
10 1010 = 10101 100021 × 2 × 2 × 1 = 10110 133100 × 302 × 22 × 1 = 2212
332200 × 302 × 11 × 1 = 2212 		≥3 30304 × 41 × 5 × 5 × 2 × 2 × 1 = 1411
30304 × 2152 × 14 × 5 = 1412
35052 × 104 × 41 × 32 = 1412
1401405 × 332 × 5 × 2 = 1412
12323504 × 1054 = 1412

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/1/09.