I'm doing a lot of reading on the unary number system, and couldn't really find any real any algorithms for which the unary system is faster than the binary system.
For example, the prime factorization problem is linear in the unary system, which is a well known fact. However, this is only a trick. As with unary we call $N$ the number we wish to factor, but in the binary problem $N$ is the number of bits. So essentially, the unary system (to adjust for binary) really goes from $O(N)$ to $O(2^N)$.
Is there any problem that is legitimately faster, when adjusting for size, for the unary numerals over the binary numerals?