Can any base ten repunit, other than 1, be a perfect cube?

If so, give an example. Otherwise prove that no base ten repunit (other than 1) can be a perfect cube.

The following table of repunits up to 50 1's long shows each one's prime factors that are under 10 million. If there is any value left after dividing by all these primes, if the value exceeds 10 million squared, the remaining value is shown after a *. (Under 10 million squared, we know it's a prime.) These values were then put through a probabilistic prime test. Some were found to be probably prime (exceedingly likely) and are marked prime; others were found definitely not to be prime and are marked not prime.

Perhaps this factorization into primes may provide some pattern by which the puzzle can be solved, as each prime factor must appear a multiple of three times, which doesn't seem likely as the numbers of prime factors get larger, but of course that's no proof of non-existence.      

N       Nth repunit
(factorization below each line)
 
1       1

2       11
11

3       111
3 37

4       1111
11 101

5       11111
41 271

6       111111
3 7 11 13 37

7       1111111
239 4649

8       11111111
11 73 101 137

9       111111111
3 3 37 333667

10      1111111111
11 41 271 9091

11      11111111111
21649 513239

12      111111111111
3 7 11 13 37 101 9901
13      1111111111111
53 79 265371653

14      11111111111111
11 239 4649 909091

15      111111111111111
3 31 37 41 271 2906161

16      1111111111111111
11 17 73 101 137 5882353

17      11111111111111111
2071723 5363222357

18      111111111111111111
3 3 7 11 13 19 37 52579 333667

19      1111111111111111111
* 1111111111111111111  prime

20      11111111111111111111
11 41 101 271 3541 9091 27961

21      111111111111111111111
3 37 43 239 1933 4649 10838689

22      1111111111111111111111
11 11 23 4093 8779 21649 513239

23      11111111111111111111111
* 11111111111111111111111 prime

24      111111111111111111111111
3 7 11 13 37 73 101 137 9901 99990001

25      1111111111111111111111111
41 271 21401 25601 182521213001

26      11111111111111111111111111
11 53 79 859 * 280846283204599997 not prime

27      111111111111111111111111111
3 3 3 37 757 333667 * 440334654777631 prime

28      1111111111111111111111111111
11 29 101 239 281 4649 909091 121499449

29      11111111111111111111111111111
3191 16763 43037 62003 77843839397

30      111111111111111111111111111111
3 7 11 13 31 37 41 211 241 271 2161 9091 2906161

31      1111111111111111111111111111111
2791 6943319 * 57336415063790604359  prime

32      11111111111111111111111111111111
11 17 73 101 137 353 449 641 1409 69857 5882353

33      111111111111111111111111111111111
3 37 67 21649 513239 * 1344628210313298373 prime

34      1111111111111111111111111111111111
11 103 4013 2071723 * 117957818840753430733  not prime

35      11111111111111111111111111111111111
41 71 239 271 4649 123551 * 102598800232111471   prime

36      111111111111111111111111111111111111
3 3 7 11 13 19 37 101 9901 52579 333667 999999000001

37      1111111111111111111111111111111111111
2028119 * 547853016076034547830334961169   not prime

38      11111111111111111111111111111111111111
11 * 1010101010101010101010101010101010101

39      111111111111111111111111111111111111111
3 37 53 79 * 239073561261285168617387389778123  not prime

40      1111111111111111111111111111111111111111
11 41 73 101 137 271 3541 9091 27961 1676321 5964848081

41      11111111111111111111111111111111111111111
83 1231 538987 * 201763709900322803748657942361  prime

42      111111111111111111111111111111111111111111
3 7 7 11 13 37 43 127 239 1933 2689 4649 459691 909091 10838689

43      1111111111111111111111111111111111111111111
173 1527791 * 4203852214522105994074156592890477   not prime

44      11111111111111111111111111111111111111111111
11 11 23 89 101 4093 8779 21649 513239 * 1112470797641561909  not prime

45      111111111111111111111111111111111111111111111
3 3 31 37 41 271 238681 333667 2906161 * 4185502830133110721 prime

46      1111111111111111111111111111111111111111111111
11 47 139 2531 * 6108857605465744444444383355868389787  not prime

47      11111111111111111111111111111111111111111111111
* 11111111111111111111111111111111111111111111111

48      111111111111111111111111111111111111111111111111
3 7 11 13 17 37 73 101 137 9901 5882353 * 999900000000999999990001 not prime

49      1111111111111111111111111111111111111111111111111
239 4649 * 1000000100000010000001000000100000010000001   not prime

50      11111111111111111111111111111111111111111111111111
11 41 251 271 5051 9091 21401 25601 * 14396532879144434243285201  not prime

Posted by Charlie on 2011-10-25 11:59:02