Is there a log-space algorithm for divisibility?

Question

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?

Answer 1

(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.

Answer 2

Edited to add: As Gadi points out in a comment, this answer is wrong.

If you have a logspace algorithm to verify $x \times y = z$, then since you're not concerned with running time, you can simply check, for all $c$ with $1 < c \le b$, whether $a \times c = b$.