11
$\begingroup$

On a mailing list (math-fun) that I subscribe to Dan Asimov asked what's the most efficient way to calculate the leading decimal digits (say 10 of them) of $(p/q)^n \bmod 1$ where $p$ and $q$ are fixed (think of $p/q = 3/2$) and $n$ varies. There were a number of suggestions, but all of them clearly had complexity proportional to $n$. So my question is, for concreteness, let $b(n) = \lfloor 2(3/2)^n \rfloor \bmod 2$ (the leading fractional bit of $(3/2)^n$). Suppose that $n$ is specified in binary. What is the complexity (both time and space) of calculating the function $b(n)$? After thinking about it for a while I wouldn't be surprised if it's exp-space hard. Does anyone know anything about this?

$\endgroup$
2
  • $\begingroup$ You may want to change that title: when I read the question on the home page, the first thing I thought is "what's the problem, the leading bit is always 1..." $\endgroup$ Jul 6, 2011 at 9:19
  • $\begingroup$ @Federico, good point. I added "fractional" before bit. $\endgroup$ Jul 6, 2011 at 13:57

1 Answer 1

4
$\begingroup$

From the review of Mika Hirvensalo, Juhani Karhumäki, and Alexander Rabinovich, Computing partial information out of intractable: powers of algebraic numbers as an example, J. Number Theory 130 (2010), no. 2, 232–253, MR2564895 (2010j:11117), it looks like there may be something of interest there.

$\endgroup$
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.