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 Repunit Rigor (Posted on 2011-10-25)
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.

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (2 votes)

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 factorizations, in case they help | Comment 1 of 5

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       1111`
`3       1113 37`
`4       111111 101`
`5       1111141 271`
`6       1111113 7 11 13 37`
`7       1111111239 4649`
`8       1111111111 73 101 137`
`9       1111111113 3 37 333667`
`10      111111111111 41 271 9091`
`11      1111111111121649 513239`
`12      1111111111113 7 11 13 37 101 990113      111111111111153 79 265371653`
`14      1111111111111111 239 4649 909091`
`15      1111111111111113 31 37 41 271 2906161`
`16      111111111111111111 17 73 101 137 5882353`
`17      111111111111111112071723 5363222357`
`18      1111111111111111113 3 7 11 13 19 37 52579 333667`
`19      1111111111111111111* 1111111111111111111  prime`
`20      1111111111111111111111 41 101 271 3541 9091 27961`
`21      1111111111111111111113 37 43 239 1933 4649 10838689`
`22      111111111111111111111111 11 23 4093 8779 21649 513239`
`23      11111111111111111111111* 11111111111111111111111 prime`
`24      1111111111111111111111113 7 11 13 37 73 101 137 9901 99990001`
`25      111111111111111111111111141 271 21401 25601 182521213001`
`26      1111111111111111111111111111 53 79 859 * 280846283204599997 not prime`
`27      1111111111111111111111111113 3 3 37 757 333667 * 440334654777631 prime`
`28      111111111111111111111111111111 29 101 239 281 4649 909091 121499449`
`29      111111111111111111111111111113191 16763 43037 62003 77843839397`
`30      1111111111111111111111111111113 7 11 13 31 37 41 211 241 271 2161 9091 2906161`
`31      11111111111111111111111111111112791 6943319 * 57336415063790604359  prime`
`32      1111111111111111111111111111111111 17 73 101 137 353 449 641 1409 69857 5882353`
`33      1111111111111111111111111111111113 37 67 21649 513239 * 1344628210313298373 prime`
`34      111111111111111111111111111111111111 103 4013 2071723 * 117957818840753430733  not prime`
`35      1111111111111111111111111111111111141 71 239 271 4649 123551 * 102598800232111471   prime`
`36      1111111111111111111111111111111111113 3 7 11 13 19 37 101 9901 52579 333667 999999000001`
`37      11111111111111111111111111111111111112028119 * 547853016076034547830334961169   not prime`
`38      1111111111111111111111111111111111111111 * 1010101010101010101010101010101010101`
`39      1111111111111111111111111111111111111113 37 53 79 * 239073561261285168617387389778123  not prime`
`40      111111111111111111111111111111111111111111 41 73 101 137 271 3541 9091 27961 1676321 5964848081`
`41      1111111111111111111111111111111111111111183 1231 538987 * 201763709900322803748657942361  prime`
`42      1111111111111111111111111111111111111111113 7 7 11 13 37 43 127 239 1933 2689 4649 459691 909091 10838689`
`43      1111111111111111111111111111111111111111111173 1527791 * 4203852214522105994074156592890477   not prime`
`44      1111111111111111111111111111111111111111111111 11 23 89 101 4093 8779 21649 513239 * 1112470797641561909  not prime`
`45      1111111111111111111111111111111111111111111113 3 31 37 41 271 238681 333667 2906161 * 4185502830133110721 prime`
`46      111111111111111111111111111111111111111111111111 47 139 2531 * 6108857605465744444444383355868389787  not prime`
`47      11111111111111111111111111111111111111111111111* 11111111111111111111111111111111111111111111111`
`48      1111111111111111111111111111111111111111111111113 7 11 13 17 37 73 101 137 9901 5882353 * 999900000000999999990001 not prime`
`49      1111111111111111111111111111111111111111111111111239 4649 * 1000000100000010000001000000100000010000001   not prime`
`50      1111111111111111111111111111111111111111111111111111 41 251 271 5051 9091 21401 25601 * 14396532879144434243285201  not prime`

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

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