Wikipedia:Reference desk/Archives/Mathematics/2022 June 22

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June 22

rep number in the most bases?

Let a number k be considered a rep in base b>1 if k in base b all has the same digits and isn't a single digit. So for example, 42 is a rep in base 4 since 42base10 = 222base4. (in would also be a rep in base 13 (33base13), base 20 (22base20) and base 41 (11base41) Since for any number k , the maximum number of bases (b>1) that it can be a rep in is k-1 (since it would be a single digit in any base > k).

Is there any way to calculate what the lowest number m where it is a rep in 10 different bases?Naraht (talk) 13:52, 22 June 2022 (UTC)

The decimal number 111111111111111111111111111111111111111111111111111111111111, consisting of 60 ones can be written as a rep in ten (or even eleven) different bases. I don't know if it is the lowest, but it is algorithmically trivial to try all smaller numbers one by one, since there are only a finite number :).  --Lambiam 15:15, 22 June 2022 (UTC)

The homographic binary number – 1152921504606846975 in decimal – also has this property, which somewhat reduces the search space.  --Lambiam 15:21, 22 June 2022 (UTC)

A search gives 336 as the smallest: it is a repdigit in bases 20, 23, 27, 41, 47, 55, 83, 111, 167 and 335. For example, 77₄₇ = 336. It is disappointingly trivial compared to 42, since it is a two-digit number in all these bases; the equation

$m=d{\frac {b^{k}-1}{b-1}}$

for a repdigit of length $k$ in base $b,$ where $d<b$, can be simplified for the case $k=2$ into $m=d(b+1).$ So all that is needed is a number that has ten factors smaller than its square root, which correspond to the digits to be repeated. If $d$ is such a factor, $b=m/d-1$ works for the base.  --Lambiam 17:01, 22 June 2022 (UTC)

So to make it interesting, we should require at least triple digits; compare the Goormaghtigh conjecture.  --Lambiam 17:31, 22 June 2022 (UTC)