UVa 583 - Prime Factors

Explanation
While a completely dumb approach to factoring a number $$n$$ is to try dividing by all possible factors $$f$$ in the range $$2 \le f \le n - 1$$ and checking their remainder, this will timeout. However, a slightly smarter approaching, trying all factors f in the range $$2 \le f \le \sqrt{n}$$ will run in time.

The reason checking the reduced set of candidates works is essentially that the smaller factor is never bigger than $$\sqrt{n}$$, a very short proof follows.

Consider a positive integer $$n$$ with positive factors $$a$$ and $$b$$, so $$n = a \times b$$. Without loss of generality, choose to label the smaller number $$a$$. So then $$a \le b$$, and $$a \times a \le a \times b \le n$$, and therefore $$a \le \sqrt{n}$$.

Gotcha's
Negative numbers are allowed, negate the input and remember the -1 coefficient before factoring.

Optimizations
These are not neccesary but will give you a faster time. The first optimization is simple, 2 is the only even prime, so after two, we can check only odd factors, which will reduce the possible factor space by a factor of 2, and give a nearly equal speed up.

An optimization that is a bit harder to implement is to pre-compute all the primes $$p$$ in the range $$2 \le p \le \sqrt{2^{31}}$$ with the Prime Sieve of Eratosthenes and use that as the candidate list.

There are only about ~5,000 primes from $$2 \le p \le \sqrt{2^{31}}$$ the time that it takes to generate all these primes are not as significant as making your program as optimized as possible on output.
 * Store the result in a string. (will save most of your time) output takes a long time.
 * While the number you are working with is not 1.

Input
-190 -191 -192 -193 -194 195 196 197 198 199 200 15152412 634637 12341 7 43 27724 0

Output
-190 = -1 x 2 x 5 x 19 -191 = -1 x 191 -192 = -1 x 2 x 2 x 2 x 2 x 2 x 2 x 3 -193 = -1 x 193 -194 = -1 x 2 x 97 195 = 3 x 5 x 13 196 = 2 x 2 x 7 x 7 197 = 197 198 = 2 x 3 x 3 x 11 199 = 199 200 = 2 x 2 x 2 x 5 x 5 15152412 = 2 x 2 x 3 x 11 x 191 x 601 634637 = 43 x 14759 12341 = 7 x 41 x 43 7 = 7 43 = 43 27724 = 2 x 2 x 29 x 239