A little bit of a Dixon.

book/dixon.tex

@@ -1,6 +1,21 @@
-\chapter{Dixon}
+\chapter{Dixon \label{chap:dixon}}
 
-dixon!
+~\cite{dixon} describes a class of ``probabilistic algorithms'' for finding a
+factor of any composite number, at a computational cost asymptotically  best
+than all other ones previously described:
+\bigO{\beta(\log N \log \log N)^{\rfrac{1}{2}}}
+for some constant $\beta > 0$.
+
+\paragraph{Kraitchick} was the first one popularizing the idea the instead of
+looking for integers $\angular{x, y}$ such that $x^2 -y^2 = N$ -recall Fermat's
+problem, formulated in equation ~\ref{eq:fermat_problem}, it is sufficient to
+look for \emph{multiples} of $N$:
+\begin{align}
+  x^2 - y^2 \equiv 0 \pmod{N}
+\end{align}
+and, once found, claim that $\gcd(N, x \pm y)$ are non-trial divisors of $N$
+just as we did in \ref{sec:fermat:implementation}.
+On the top of this,
 
 %%% Local Variables:
 %%% mode: latex

book/fermat.tex

@@ -49,7 +49,7 @@ below here to better clarify the limits of this algorithm.
   of this factorization method approaches \bigO{N}.
 \end{remark}
 
-\section{An Implementation Perspective}
+\section{An Implementation Perspective \label{sec:fermat:implementation}}
 
 At each iteration, the $i-$th state is hold by the pair $\angular{x, x^2}$.\\
 The later step, described by $\angular{x+1, (x+1)^2}$ can be computed efficiently

book/library.bib

@@ -126,4 +126,16 @@
   language = {English},
   year = 1982,
   publisher = {American Mathematical Society},
-}
+}
+
+@article{dixon,
+  title = {Asymptotically Fast Factorization of Integers},
+  author = {Dixon, John D.},
+  journal = {Mathematics of Computation},
+  volume = {36},
+  number = {153},
+  pages = {pp. 255-260},
+  url = {http://www.jstor.org/stable/2007743},
+  ISSN = {00255718},
+  year = {1981}
+}

book/question_authority.tex

@@ -158,6 +158,7 @@
 \include{pollard-1}
 \include{pollard+1}
 \include{pollardrho}
+\include{dixon}
 
 \include{conclusions}