Speeding up my implementation of Project Euler #3
By inspection, n has no even factors.  You only need to try odd factors.

Whenever you find an i that is a divisor of n, you should factor out all powers of i.  There is also no need for i to start from scratch — your hunt for divisors should proceed monotonically upward.

Your search can end when i reaches or surpasses the square root of n.  At that point, n is the answer to the challenge.

(A reasonable algorithm should obtain the answer within milliseconds.)