[seqfan] What I've found so far about natural numbers not sharing digits with their squares

WayneMV waynemv at gmail.com
Tue Oct 16 04:24:57 CEST 2018

I've been investigating integers that do not share any (base 10) digits in common with their squares, or higher powers. I haven't yet generated an interesting enough (in my opinion) sequences from this yet, but if I come up with any I will submit them. I was working on this myself for a couple weeks before seeing another post in the group recently that mentioned a similar property. My only questions are general ones: What more can be said? And what can be proved? Also, who might want to help me work on optimizing my search methods to be more efficient? (If so, private message me.)

The following is the highest example I have so far found where N has five distinct digits. But my brute force search method wasn't very systematic, so I suspect higher examples could be found with only a little more effort:


An example where N has three distinct digits:


An example where N has only two distinct digits:


My original question would have been whether this can be proven to be any upper limit to how large of N can work. But I started on this I realized that the square of an arbitrarily long series of 3s easily provably only ever contains the digits 1,0,8, and 9:

N  =33333333333333333333333333333333333333333333333

And the last digit can be 4 or 8:


Other minor modifications to the series of 3s are likely possible.

QUESTION: Might there be a maximum value that works for N vs N^2 in the cases where N doesn't contain any 3s? (That is, after some point, does every working value of N contain at least one 3? Or might it be shown that some pattern of 2s, 4s, and 9s continues arbitrarily long?)

I also haven't found any regular patterns for cubes, that would guarantee that arbitrarily large values work. The largest cube I have so far found that works is:


For 7th and 11th powers, these may well be the maximum possible values that work. I've not found any higher working values under 5 million. 


QUESTION: Do any of you know a method for actually proving that conjecture?

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