This is probably one of the most important environments to implement! This is because it's required to show syntax to the reader who wishes to learn about \TeX\ and \LaTeX.


\begin{verbatim}
 begin{verbatim}
  some \LaTeX\ here will not get parsed!
 end{verbatim}
\end{verbatim}

which looks like this when rendered:

\begin{verbatim}
some \LaTeX\ here will not get parsed!
\end{verbatim}

\begin{example}
This method based on the Chinese Remainder Theorem, is used to factor a large number N.


Pick a random integer for a starting number  $ = y_0 = z_0 $ Then recursively apply the function $f(x) = x^2 + a \bmod N$  twice to $z$, and once to $y$, where $a$ is the random remainder to start with.

$$y_1 = f(y_0)$$
$$z_1 = f(f(y_0))$$

We use this process to generate remainders until we hit a loop and one of them is divisible by $p$, where $N=pq$. which has a good chance of being achieved by $\sqrt{p}$ factors. 


We test this at each iteration by taking the greatest common divisor of the difference between the values and the number you are trying to factor.

$$\gcd(| y_k-x_k| , N)$$

If the $\gcd$ 1, then find $x_k $ and $ y_k$ and recursively apply the algorithm. If you do not get a one, then the number is a factor of $N$.

An algorithm written in Scheme emulates this procedure:

\begin{verbatim}
(define (pollards n y z fn)
        (let* ((n-y (fn y))
                  (n-z (fn (fn z)))
                  (g (gcd (abs (- n-y n-z)) n)))
            (if (= g 1) (pollards n n-y n-z fn) g)))
                    
\end{verbatim}
\end{example}

Here is an example of a program written in an assembly language called MIPS:

\begin{example}
This emulates {\tt *x = *y } where {\tt x} and {\tt y} are pointers
\begin{verbatim}
        .data
x:      .word 4
y:      .word 5

        .text
main:
    la      $s0, x
    la      $s1, y
    lw      $t1, 0($s1) # read y into temp var
    sw      $t1, 0($s0) # store into x
    lw      $t1, 0($s0) # read from x into temp var
    move    $a0, $t1
    li $v0, 1
    syscall			
    li      $v0, 10		
    syscall			
\end{verbatim}
\end{example}