# Integer factorization: comparing with floor

While working on integer factorization algorithm I came to the next problem:

$$\frac{a}{ex} = \lfloor{\frac{a}{ex}\rfloor} + c$$

• $a$ the number I want to factor
• $x$ factor of $a$
• $e$ positive integer that I choose

For $e = 1$ I found a linear representation for short sequences of $\lfloor{\frac{a}{x}\rfloor}$, this allows me to compare it to $\frac{a}{x}$ and find the value of $x$. The problem is that there are to many sequences, for this to be efficient. So I decrease their amount by using $e > 1$. Unfortunately it created $c$, and now I have to test all its possible values, it looks some thing like this:

• for $e=2$, $c$ can be $\frac{1}{2}$
• for $e=3$, $c$ can be $\frac{1}{3}$ or $\frac{2}{3}$
• for $e=4$, $c$ can be $\frac{1}{4}$ or $\frac{2}{4}$ or $\frac{3}{4}$

Is there any way that I can predict the values of $c$, or give it a better estimation?

The value of $c$ is given by the formula $$c = \frac{(a/x) \pmod{e}}{e}.$$
• $c$ is the bottle neck of my algorithm, I want to reduce the values it may have. When factoring large numbers like 1024 bit, $e$ have to be some where at 500+ bits, it makes it very unclear to what $c$ might be. – Ilya Gazman Jan 28 '15 at 20:14