What's Special About 7658?

I can thank Cliff Pickover a tweet for identifying what's special about the number 7658. See Figure 1 which is a screenshot of his tweet.

Figure 1: link

So 7658 is the largest number with distinct digits that doesn't have any digits in common with its cube. 

\(7658^3 =     449103134312\)

Here is a table of all numbers with the property that there are no digits in common (permalink).

number   cube

  2        8

  3        27

  7        343

  8        512

  27       19683

  43       79507

  47       103823

  48       110592

  52       140608

  53       148877

  63       250047

  68       314432

  92       778688

  157      3869893

  172      5088448

  187      6539203

  192      7077888

  263      18191447

  378      54010152

  408      67917312

  423      75686967

  458      96071912

  468      102503232

  478      109215352

  487      115501303

  527      146363183

  587      202262003

  608      224755712

  648      272097792

  692      331373888

  823      557441767

  843      599077107

  918      773620632

  1457     3092990993

  1587     3996969003

  1592     4034866688

  4657     100999381393

  4732     105958111168

  5692     184414333888

  6058     222324747112

  6378     259449922152

  7658     449103134312

This fact got me interested in finding out the largest number with distinct digits that has no digits in common with its square. Checking in the range up to one million, which is about the limit for the online SageMathCell, I found the number to be 639172 whose square is    408540845584 (permalink). I strongly suspect that this is the largest number. So we have:

\(639172^2 =   408540845584\)

What about fourth powers? What is the largest number that, when raised to the fourth power, has no digits in common with the base number? That number turns out to be 2673.

\(2673^4 = 51050010415041\)

There doesn't appear to be any numbers satisfying the fifth power but 92 is the largest such number when sixth powers are involved:

\(92^6 =  606355001344\)

I'll leave off there. So, in summary, our investigation into what is special about 7658 led us to discover some associated numbers (92, 2673 and 639172) that are the largest possible numbers when powers of 6, 4 and 2 are considered.