A Curious Property of 82000
The number 82000
in base 10 is equal to 10100000001010000 in base 2, 11011111001 in base 3, 110001100 in base 4, and 10111000 in base 5. It is the smallest integer bigger than 1 whose expressions in bases 2, 3, 4, and 5 all consist entirely of zeros and ones.
What is remarkable about this property is how much the situation changes if we alter the question slightly. The smallest number bigger than 1 whose base 2, 3, and 4 representations consist of zeros and ones is 4. If we ask the same question for bases up to 3, the answer is 3, and for bases up to 2, the answer is 2. The question does not make sense for base 1, which is what leads to the sequence in the picture: [undefined], 2, 3, 4, 82000.
The graphic comes from a blog post by Thomas Oléron Evans
. Most of the post discusses the intriguing problem of finding the next term in this sequence, and whether the next term even exists. In other words, does there exist an integer greater than 1 whose representations in bases 2, 3, 4, 5, and 6 all consist entirely of zeros and ones?
The number 82000 does not satisfy these conditions, because the representation of this number in base 6 is 1431344. This means that the next number in the sequence, if it exists, must be some number bigger than 82000 whose representations in bases 2, 3, 4, and 5 all consist entirely of zeros and ones. Unfortunately, even these weaker conditions are very difficult to satisfy. An exhaustive search has been carried out up to 3125 digits in base 5 and no solution exists in this range.
The upshot of this is that, if the next term in the sequence exists, it must have more than 2184 digits in base 10. (The 2184 comes from multiplying 3125 by the base 10 logarithm of 5.) However, there is also no known proof that the next term in the sequence does not exist.Relevant links
Thomas Oléron Evans's blog post has much more discussion of this problem, at http://www.mathistopheles.co.uk/maths/covering-all-the-bases/solution-covering-all-the-bases/
Details of the exhaustive search can be found in the notes to the sequence http://oeis.org/A146025
in the On-Line Encyclopedia of Integer Sequences
There is a nice online number base converter tool at http://www.cleavebooks.co.uk/scol/calnumba.htm#mathematics #sciencesunday