Problem of the Month (March 2008)

There are only a few positive integers A where A2 contains exactly 2 different non-zero digits. You can find them on many web sites, and should have no problem finding them for yourself. Similarly, there seem to be only a few A where A3 contains exactly 3 different non-zero digits. What is the largest one? In general, what is the largest nth power that contains exactly n different non-zero digits?

This month we investigate other patterns involving nth powers that give rise to numbers with exactly n different non-zero digits. For example, for what positive integers A and B do A + B and A2 + B2 both contain the same two digits? The largest non-trivial example I've managed to find is 82741 + 144486 = 227227 and 827412 + 1444862 = 27722277277. Are there larger examples? What if we allow the digits in A + B to be different than the digits in A2 + B2? What about higher powers? What other patterns give surprising results?

ANSWERS

The following people sent solutions this month: Giovanni Resta, Luke Pebody, Yves Hester, Claudio Baiocchi, and George Sicherman.

Here are the largest known nth powers with n digits:

FormNumber
of Digits		Largest Known Example
A22816192 = 6661661161
A331467963 = 3163316636166336
A44847594 = 51611121555566262561
A5521143255 = 42253114333245121312442314453125
(Giovanni Resta)
A6645295338616 = 8636167634317137668616 844443347687331783434473474447817161
(Giovanni Resta)
A772897061580967 =
1712789877859251998896815797915181677696 18276828525177119221996256516887927586816
(Giovanni Resta)
A887164570530948 =
69425864962262381253935889556533538935158818438 832114326146546514844981265541243248464849346816
(Giovanni Resta)
A9911954349912329 =
498578646328835566488514727474191378397748897885953663 8328432494741295723697842777912875263643474151585349632
(Giovanni Resta)

Here are the largest known examples for sums of nth powers with n digits:

FormNumber
of Digits		Largest Known Example
A + B
A2 + B2 		same 2183425228501 + 438841438125 = 622266666626
1834252285012 + 4388414381252 = 226226622266262266222626
(Giovanni Resta)
A + B + C
A2 + B2 + C2
A3 + B3 + C3 		same 32675 + 12021 + 21440 = 36136
26752 + 120212 + 214402 = 611333666
26753 + 120213 + 214403 = 11611631166136
(Giovanni Resta)
A + B + C + D
A2 + B2 + C2 + D2
A3 + B3 + C3+ D3
A4 + B4 + C4 + D4 		same 446 + 271 + 386 + 1224 = 1927
462 + 2712 + 3862 + 12242 = 1722729
463 + 2713 + 3863 + 12243 = 1911279727
464 + 2714 + 3864 + 12244 = 2272129192929
(Giovanni Resta)

FormNumber
of Digits		Largest Known Example
A + B
A2 + B2 		any 2Infinitely Many!
[3]2[3]1 + [3]4[3]1 = [6]6[6]2
[3]2[3]12 + [3]4[3]12 = [2][2]21[1][1]22
(Giovanni Resta)
A + B + C
A2 + B2 + C2
A3 + B3 + C3 		any 36421 + 17731 + 42497 = 66649
64212 + 177312 + 424972 = 2161612611
64213 + 177313 + 424973 = 82588522882825
(Giovanni Resta)
A + B + C + D
A2 + B2 + C2 + D2
A3 + B3 + C3+ D3
A4 + B4 + C4 + D4 		any 4487 + 833 + 1198 + 9777 = 12295
4872 + 8332 + 11982 + 97772 = 97955991
4873 + 8333 + 11983 + 97773 = 936993665665
4874 + 8334 + 11984 + 97774 = 9139993831893939
(Giovanni Resta)

Here are the largest known examples for sums of nth powers with 2 digits:

FormNumber
of Digits		Largest Known Example
A3 + B3 22489753 + 22884463 =
11999991191911999911
(Giovanni Resta)
A4 + B4 + C4 283764 + 261394 + 571424 =
11133333313313133313
(Giovanni Resta)
A5 + B5 + C5 + D5 223675 + 27635 + 38605 + 41225 =
2282222888228228282
(Giovanni Resta)
A6 + B6 + C6 + D6 + E6 21256 + 1596 + 4226 + 4536 + 14606 =
9699699696696969699
(Giovanni Resta)
A7 + B7 + C7 + D7 + E7 + F7 21387 + 1567 + 1967 + 3187 + 3287 + 3457 =
1333333133333331113
(Giovanni Resta)
A8 + B8 + C8 + D8 + E8 + F8 + G8 2438 + 578 + 578 + 728 + 938 + 938 + 1808 =
1114144444444111141
(Giovanni Resta)
A9 + B9 + C9 + D9 + E9 + F9 + G9 + H9 2149 + 239 + 449 + 569 + 919 + 959 + 1069 + 1169 =
6556655556656655665
(Giovanni Resta)

Here are the largest known examples for some other symmetrical patterns:

FormNumber
of Digits		Largest Known Example
A2 + B
A + B2 		same 2
distinct numbers 		105662 + 25810 = 111666166
10566 + 258102 = 666166666
(George Sicherman)
A2 + B + C
A + B2 + C
A + B + C2 		same 2
distinct numbers 		1402 + 161 + 3161 = 22922
140 + 1612 + 3161 = 29222
140 + 161 + 31612 = 9992222
(Giovanni Resta)
A2 + B2 + C
A2 + B + C2
A + B2 + C2 		same 2
distinct numbers 		482 + 532 + 1053 = 6166
482 + 53 + 10532 = 1111166
48 + 532 + 10532 = 1111666
(Giovanni Resta)

FormNumber
of Digits		Largest Known Example
A2 + B
A + B2 		any 2
distinct numbers 		277902 + 438177 = 772722277
27790 + 4381772 = 191999111119
(Giovanni Resta)
A2 + B + C
A + B2 + C
A + B + C2 		any 2
distinct numbers 		392 + 63 + 11547 = 13131
39 + 632 + 11547 = 15555
39 + 63 + 115472 = 133333311
(Giovanni Resta)
A2 + B2 + C
A2 + B + C2
A + B2 + C2 		any 2
distinct numbers 		1532 + 21532 + 7848 = 4666666
1532 + 2153 + 78482 = 61616666
153 + 21532 + 78482 = 66226666
(Giovanni Resta)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/12/08.