Why 82,000 is an extraordinary number - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users cite it as the origin of the 82,000 puzzle when asking follow-up questions about numbers written with only 0s and 1s in several bases, including negative bases.
Recommended by 1 people across 1 different months, 2015 to 2023.
What it is
A short Numberphile video in which James Grime explains why 82,000 is special: it can be written using only the digits 0 and 1 in bases 2, 3, 4 and 5. He notes it is unknown whether any larger number does this, and it follows from an exploration by Thomas Oléron Evans.
What people said
Recommends EDS · 2023-04 · Mathematics Stack Exchange · on “Examples of patterns that eventually fail”
The below two patters fail early on and nobody who's spent some time on basic maths would think they would continue. But considering they're so easily stated, I feel they're still in the spirit of the question for a truly lay audience. The smallest integer greater than one that can be written entirely in ones and zeros in base $2$ is $2$ : $$2=(1)\cdot 2^1+(0)2^0$$ The smallest integer greater than one that can be written entirely in ones and zeros in bases $2$ and $3$ is $3$ : $$3=(1)3^1 + (0)3^0=(1)2^1+(1)2^0$$ The smallest integer greater than one that can be written entirely in ones and zeros in bases $2$ , $3$ , and $4$ is $4$ : $$4=(1)4^1+(0)4^0=(1)3^1+(1)3^0=(1)2^2+(0)2^1+(0)2^0$$ The
Cites Jed · 2015-10 · Mathematics Stack Exchange · on “Numbers Written With Only 1s and 0s”
I just watched this video [this video] that talked about the smallest number that can be written in some base(s) using only the digits 1 and 0. Afterwards, I wondered if there was a way to find the number of integers that can be written using only 1s and 0s in bases 3 and also base 4. I am not sure how to start on this problem. Hints and solutions are appreciated.
Cites gilianzz · 2016-08 · Mathematics Stack Exchange · on “Numberphile's "Why 82000 is an extraordinary number" applied to negative bases?”
In a numberphile video [this video] it is shown why 82000 is an extraordinary number. It can be written using only zeros and ones in base 2, base 3, base 4 and base 5. It is unknown if there are larger numbers for which this is true, and if it can apply to base 6 and higher. $82000_{10}=10100000001010000_2=11011111001_3=110001100_4=10111000_5$ My question is: is there a similar "special" number that can be written using only zeros and ones in decreasingly negative bases (base -2, base -3, base -4, etc)? Using python I found that from -100000000 to 100000000 the most negative number that can be written in base -3 and base -4 is $-196335$, the most positive one is $4353$.