The second part of Pollard *rho* factorization method concerns detection of the fact that a sequence has become
periodic. Pollard's original suggestion was to use the idea attributed to Floyd of
comparing
to
for all .
Brent's improvement to Pollard's method concerns how to detect periodicity, and replaces
Floyd's method with the following algorithm. Keep only one running copy of . If is a power of a base , let , and at each step, compare the current value with the saved value . In the factorization case, instead of comparing with , compute

More generally, Brent (1980) considered using any base for saving values instead of . However, he found to be very close to optimal.