Problem of the Month (October 2004)

This month we investigate square numbers that have repeating (non-overlapping, not all zero) blocks of digits.

1) What is the smallest square that starts and ends with the same n digits?
For example, 162622 = 2644 5 2644 starts and ends with the same 4 digits.

2) What is the smallest square that contains c copies of the same block of n digits?
For example, 2501252 = 625 625 15 625 contains 3 copies of the block 625.

3) What squares contain cn digits, and when considered as c blocks of n digits, contain only 2 different blocks?
For example, 7277272 = 529 586 586 529 contains only the blocks 529 and 586.

4) What are the answers to these questions in other bases?

5) What are the answers to these questions for higher powers?


ANSWERS

1)

Joseph DeVincentis, Philippe Fondanaiche, Patrick De Geest, and Richard Sabey were interested in when a square could be 2 identical blocks of n digits. This can clearly only occur when 10n + 1 has a square factor. This happens for n=11, 21, 33, 39, ....

Richard Sabey sent the most detailed analysis: Pick any prime p, with certain exceptions. Find the period 2n of the decimal (or, generalising, base-b) expansion of 1/p2, thus proving that p2 divides b2n-1 = (b^n+1)(bn-1). If the period of the expansion of 1/p doesn't divide n, that is, p doesn't divide bn-1, then p2 divides bn+1. Then N = a(bn+1)/p where a is chosen so as to make N2/(bn+1) an n-digit number. From this, the primitive solution using prime p and base b, more can be found by replacing n by any odd multiple of n.

Patrick De Geest noted that 363636363642 = 13223140496 13223140496 and 636363636372 = 40495867769 40495867769 and 36363636364 + 63636363637 = 100000000001.

Here are the smallest squares that begin and end with the same n digits. These were all found by Luke Pebody. Evgeni Lukin and Patrick De Geest also sent some solutions.

nsmallest square
1112 =
1 2 1
21732 =
29 9 29
334892 =
121 73 121
4162622 =
2644 5 2644
519337442 =
37536 7 37536
624458652 =
598225 5 598225
7389810392 =
1519521 40 1519521
81127919552 =
12722025 1 12722025
915801780162 =
249696256 2 249696256
10105789373812 =
1119139161 1 1119139161
11363636363642 =
13223140496 13223140496
1210106909617952 =
102149622025 4 102149622025
13144517890074872 =
2088542055169 2 2088542055169
141045014636040862 =
10920555895396 1 10920555895396
1512428442688970552 =
154466187673025 5 154466187673025
16117731012579253792 =
1386059132293641 4 1386059132293641
171818833537908607012 =
33081554386211401 7 33081554386211401
1818181818151818181822 =
330578511305785124 9 330578511305785124
19116783321167882711682 =
1363834410300084224 5 1363834410300084224
201020408163165306122452 =
10412328194543940025 1 10412328194543940025
214285714285714285714292 =
183673469387755102041 183673469387755102041
22129625907602504912775972 =
1680287592177314094409 4 1680287592177314094409
231176475588235294122647062 =
13840948097135813266436 5 13840948097135813266436
2411692132722751518389559912 =
136705967606436834792081 1 136705967606436834792081
25180574095433382109784070622 =
3260700394158418971471844 8 3260700394158418971471844
261025774358974425641025774362 =
10522130355293938376334096 4 10522130355293938376334096
2715728432157284271572892715732 =
247383578126293964945894329 5 247383578126293964945894329
28114772688675047136685508849232 =
1317277006569929326388715929 5 1317277006569929326388715929
291060118543893988125610401187442 =
11238513271079096281620137536 8 11238513271079096281620137536
3018181818181818161818181818181822 =
330578512396693487603305785124 4 330578512396693487603305785124
31196073257238289854627986413692662 =
3844472220403258487071369378756 5 3844472220403258487071369378756
321024019973497148190390676334011012 =
10486169061211000822488748012201 1 10486169061211000822488748012201
333636363636363636363636363636363642 =
132231404958677685950413223140496 132231404958677685950413223140496
34108521107896646010810871712624087692 =
1177683085911548516463360048095361 4 1177683085911548516463360048095361
351194044780623280565163586475082031802 =
14257429381336982110065392162112400 5 14257429381336982110065392162112400
3610601474987179322888199721882229486012 =
112391271903788824396724054687857201 4 112391271903788824396724054687857201
37327143534347423338528973448153316644782 =
1070228920653237536415964465967012484 10 1070228920653237536415964465967012484
381000826874057771053725719656990194213442 =
10016544318362495222193336100602766336 4 10016544318362495222193336100602766336
393846153846153846153846153846153846153852 =
147928994082840236686390532544378698225 147928994082840236686390532544378698225
40181818181818181818184818181818181818181822 =
3305785123966942148869421487603305785124 9 3305785123966942148869421487603305785124
411028172305794360832305827027505605827700842 =
10571382904024926415382104115050805367056 2 10571382904024926415382104115050805367056
4211264591734137449939635965290228663737885492 =
126891026936797761663743079640747363525401 4 126891026936797761663743079640747363525401
43100089990995899910008999100089990500899910012 =
1001800629755932505941987043196480260982001 5 1001800629755932505941987043196480260982001
441000816808006931642654866788461860625970903412 =
10016342831891834729521288974571159315496281 1 10016342831891834729521288974571159315496281
4518515792009855643725399795121988572975772181612 =
342834553752234098588349159884722439388221921 8 342834553752234098588349159884722439388221921
46117906857368445457074143758872849926693418051182 =
1390202701450294077483346798830190622690993924 9 1390202701450294077483346798830190622690993924
471013406484388980895916053428175174677823022421752 =
10269927026016337802982214649939562032348730625 5 10269927026016337802982214649939562032348730625
4812660250510624695130423897972839629111976175096802 =
160281942991772853787477585659637092124893702400 1 160281942991772853787477585659637092124893702400
49103607038534572807498274789774111876858972493676002 =
1073441843390445464624896867372439658599929760000 2 1073441843390445464624896867372439658599929760000
501003236259288025889967637607119741100323625928802592 =
10064829919502311137755156813606913390517511907081 4 10064829919502311137755156813606913390517511907081

2)

Philippe Fondanaiche sent some solutions. Here are the smallest squares with c equal blocks of length n:

c \ n1234
112 =
1
42 =
16
102 =
100
322 =
1024
2112 =
1 2 1
1732 =
29 9 29
10022 =
1 004 004
100022 =
1 0004 0004
3382 =
1 4 4 4
15572 =
24 24 24 9
2501252 =
625 625 15 625
31850122 =
1 0144 3 0144 0144
42122 =
4 4 9 4 4
402042 =
16 16 36 16 16
37550102 =
14 100 100 100 100
525382 =
6 4 4 1 4 4 4
668882 =
4 7 4 4 4 5 4 4
7665922 =
4 4 3 4 4 9 4 4 6 4
82107712 =
4 4 4 2 4 4 1 4 4 4 1
910550412 =
1 1 1 3 1 1 1 5 1 1 68 1
1047140452 =
2 2 2 2 2 2 2 0 2 6 2 0 2 5
11349645852 =
1 2 2 2 5 2 2 2 04 2 2 2 2 2 5

3)

Here are the smallest squares with only m blocks of length n, 2 of which are unique.

n / m45678910
11 4 4 4
7 7 4 4
1 1 8 8 1
2 9 9 2 9
4 4 9 4 4
5 5 2 2 5
6 9 6 9 6
9 6 9 6 9 9 66 6 6 1 6 6 1 1 6 1
225 25 06 25
45 45 45 64
80 80 21 21
82 82 82 01
97 97 04 04
16 16 36 16 16
29 29 29 91 29
36 36 81 36 36
3100 100 100 996
104 104 313 104
141 001 001 001
165 165 836 836
231 625 625 625
352 352 649 649
456 456 979 456
529 586 586 529
564 004 004 004
997 997 004 004
100 100 225 100 100
46100 8299 8299 6100
7876 3626 3626 7876
8121 4361 4361 8121
(Philippe Fondanaiche)
1296 1296 2916 1296 1296
(Patrick De Geest)
540496 21487 21487 40496
(Philippe Fondanaiche)
6453289 359866 359866 453289
(Philippe Fondanaiche)

4)

Scott Reynolds found the smallest solutions for other bases. His data can be found here.

5)

Richard Sabey found that 73 = 111 and 143 = 888 in base 18. These appear to be the only rep-digit powers in other bases.

Here are the smallest cubes that begin and end with the same n digits.

nsmallest cube
173 = 3 4 3
21083 = 12 597 12
33353 = 375 95 375
466673 = 2963 4074 2963
51046363 = 11456 273880 11456
63333353 = 370375 92595 370375
745046253 = 9140625 762236 9140625
8705857363 = 35168256 83571650 35168256

Here are the smallest cubes with c equal blocks of length n:

c \ n1234
113 =
1
33 =
27
53 =
125
103 =
1000
273 =
3 4 3
1013 =
10 30 30 1
3353 =
375 95 375
34543 =
41 2066 2066 4
3363 =
4 6 6 5 6
2113 =
93 93 93 1
235313 =
1 302 9 302 0 302 91
41063 =
1 1 9 1 0 1 6
134643 =
2 44 07 44 44 13 44
23319633 =
126813 347 347 1 347 3 347
56123 =
2 2 9 2 2 09 2 8
4171383 =
72 583 72 68 72 6 72 0 72
610413 =
1 1 28 1 1 1 92 1
781213 =
5 3 5 5 8 5 1 5 5 5 61
8134643 =
2 4 4 07 4 4 4 4 13 4 4
9999993 =
9 9 9 9 700002 9 9 9 9 9
104268593 =
7 7 7 7 7 38329 7 7 5 7 7 7 9
119999993 =
9 9 9 9 9 7000002 9 9 9 9 9 9
1228115743 =
2 2 2 2 5347 2 73 2 2 2 2 2 7 2 2 4
1348361783 =
1 1 3 1 1 1 5 1 8 1 1 8 1 4 1 1 1 1 752

Another problem concerning patterns in squares can be found here.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/31/04.