This may sound like an absurd question [and in some sense it is!], but this came out of a recent mathematical riddle on The Riddler, asking for the largest number one could write with ten symbols. The difficulty with this riddle is the definition of a symbol, as the collection of available symbols is a very relative concept. For instance, if one takes the symbols available on a basic pocket calculator, besides the 10 digits and the decimal point, there should be the four basic operations plus square root and square,which means that presumably 999999999² is the largest one can on a cell phone, there are already many more operations, for instance my phone includes the factorial operator and hence 9!!!!!!!!! is a good guess. While moving to a computer the problem becomes somewhat meaningless, both because there are very few software that handle infinite precision computing and hence very large numbers are not achievable without additional coding, and because it very much depends on the environment and on which numbers and symbols are already coded in the local language. As illustrated by this X validated answer, this link from The Riddler, and the xkcd entry below. (The solution provided by The Riddler itself is not particularly relevant as it relies on a particular collection of symbols, which mean Rado’s number BB(9999!) is also a solution within the right referential.)
Archive for factorial
Le Monde weekend puzzle for the past week (I have not peeked at the solution yet!) was quite straightforward: find those n‘s such that 2n divides n! and those n‘s for which 2n-1 divides n. Looking at the problem in the plane to Montpellier, I think that the solution is that no positive n exists such that 2n divides n! and powers of 2 are those numbers for which 2n-1 divides n.
My reasoning is
- that the numbers with the highest potential is a power of 2,
- that 2n-1 divides n! when n is of the form 2m, and
- therefore that no integer n can satisfy the harder constraint.
Proving that 2n-1 divides n! when n is of the form n=2m can be done by induction: it works for n=2 and if it works for 2m, then it works for n=2m+1 by considering a separation of n! into
and by using the induction assumption that (2m)! can be divided by
Recycling the dividers for the second part leads to its being divisible by
because the very last term in the factorial is by 2m+1, which can be divided by 2m+1… Proving that integers other than the 2m‘s cannot be divided by 2n-1 again works by an induction proof on m.