TITLE: Is there a log-space algorithm for divisibility?
QUESTION [6 upvotes]: Is there an algorithm to test divisibility in space $O(\log n)$, or even in space $O(\log(n)^k)$ for some $k$?  Given a pair of integers $(a, b)$, the algorithm should return TRUE if $b$ is divisible by $a$, and FALSE otherwise.  I understand that there is no proof that divisibility is not in $L$ since that would imply $P \ne L$ which is open.  Also I understand that if there is a nondeterministic algorithm in $O(\log(n)^k)$ then there is a deterministic algorithm in $O(\log(n)^{2k})$, by Savitch's theorem.  I believe I've figured out an $L$ algorithm to verify $a*b=c$, and also an $FL$ algorithm to compute $c$ (essentially the method I was taught in grade school, reusing ink when possible), but I haven't been able to adapt it to divisibility.  Is such an algorithm for divisibility known?

REPLY [7 votes]: (This is an updated version of my comment on the question.)
Beame, Cook, and Hoover [BCH86] showed that integer divisibility is in L.  More recently, Chiu, Davida, and Litow [CDL01] showed that integer division is also in L.
References
[BCH86] Paul W. Beame, Stephen A. Cook, and H. James Hoover.  Log depth circuits for division and related problems.  SIAM Journal on Computing, 15(4):994–1003, Nov. 1986.  DOI: 10.1137/0215070
[CDL01] Andrew Chiu, George Davida, and Bruce Litow.  Division in logspace-uniform NC1.  Theoretical Informatics and Applications, 35(3):259–275, May 2001.  DOI: 10.1051/ita:2001119.