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%     This is a sample LaTeX file. Lines beginning with a percent sign are comments 
%     and will be ignored when the file is processed.
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%     The file begins with a declaration of the document class. We use the 
%     American Mathematical Society article class called ``amsart'' and
%     a twelve point typeface.
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%     Most TeX commands begin with a backslash.
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\documentclass[12pt]{amsart}
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%     Now we put in some information about the document.
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\author{Spike Specious}
\title{Crypto Homework Number 1}
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%     We are ready to begin the document 
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\begin{document}
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%     The maketitle command creates the title heading for the document 
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\maketitle
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%     Now we enter two paragraphs of ordinary text.
%     We indicate that we are starting a new paragraph by leaving a 
%     blank line.  
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      This is a test.
      This is only a test. Had this
      been a real document
      there might have been something
      interesting here. Move along now,
      move along.

      In the last paragraph I made the lines in the input file each rather short. In this paragraph I am making the lines 
      extremely long. Note that this makes no difference at all to the appearance of the processed document. 
%
%
%     For the purposes of writing homework solutions it is nice to be able to generate numbered lists.
%     Here is how you do that.
%

\begin{enumerate}

\item  Here is the first item in my numbered list.


\item  Here is the second item. It contains another numbered list, which I have indented here in
       the source file to 
       make the structure more obvious.

       \begin{enumerate}

       \item Item one.

       \item Item two.

       \end{enumerate}

\end{enumerate}

   Of course the main point is to be able to typeset some mathematics. Here is another list
   (not numbered this time) where each item is a mathematical equation or expression.
   Comparing the source code and the processed document you should be able to see how
   each formula was coded.  Some of the examples will look funny when processed as I 
   introduced some instructive mistakes in the source code. 

\begin{itemize}

%
%    Mathematical formulae begin and end with a dollar sign ($).
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\item  $2 + 2 = 4$.

\item  $2^2 = 4$.

% 
%   Parentheses { and } are used to form ``groups''. This is particularly important when 
%   you are doing superscripts and subscripts.


\item  $2^{1 + 1} = 4$, if you leave out the parentheses you get $2^1 + 1 = 4$ which is not what
    you want.


\item $c_0 = 1, c_1 = 1, c_{n+2} = c_{n+1} + c_n$.

\item $\sum_{i+1}^n i = \frac { i (i+1) } {2}$. That last equation looked a
    little squished so we may prefer 
  $\displaystyle \sum_{i+1}^n i = \frac { i (i+1) } {2}$. 



\end{itemize}




    Here is a slightly  more serious example, this time given as a displayed formula rather than
    as part of the text.
\[
    \mathrm{Ai}(x) \simeq \frac{ e^{ - \frac{2}{3} x^{3/2} } }{2 \sqrt{\pi} x^{\frac{1}{4}} }  
\]

%
%   Bracketing the formula between \[ and \] instead of two $'s  makes it into a displayed formula. 
%   \mathrm{Ai} puts the Ai into roman letters instead of italics. 
%   \pi gives the greek letter pi, and \sqrt is the square root operator
%



%
\end{document}
