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\begin{center}\framebox{\parbox{\boxlength}{\bf
CS 681: Computational Number Theory and Algebra \hfill 
Lecture 19: Smooth Numbers
\\
Lecturer: Manindra Agrawal
\hfill
Notes by: Arun Iyer
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September 20, 2005.
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\vspace{5mm}

\section{Smooth Numbers}
\begin{definition}
A number x is said to be y-smooth if all its prime factors are less than or equal to y.
\end{definition}
Example of Smooth Numbers :\\1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... are 2-smooth\\1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, ... are 3-smooth\\1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, ... are 5-smooth\\Smooth Numbers are used in Elliptic Curve Factoring, Quadratic Sieve and Number Field Sieve, the three most popular integer factoring algorithms. They are also used in index calculus method for discrete log problem. Before delving any further, we need to first get an idea of density of smooth numbers.

\section{Density of Smooth Numbers}
Let $\Psi (x,y) = \{m \leq x \, | \, m \; is \; y-smooth\}$\\Let $\psi (x,y) = |\Psi (x,y)|$\\Suppose m is y-smooth.\\Then $m=p_1^{\alpha_1} p_2^{\alpha_2} \ldots p_t^{\alpha_t}$\\where $p_1,p_2,\ldots,p_t$ are all primes less than or equal to y.

\subsection{Upper Bound on the Density}
One way to go about finding the upper bound is,
\begin{center}\begin{tabular}{lcl}
$\psi(x,y)$ & $=$ & $\prod_{i=1}^{t}(1+\log_{p_i}(x))$ \\
 & $\leq$ & $\prod_{i=1}^{t}(1+\log_{2}(x))$\\
 & $=$ & $O((\log x)^t)$\\
 & $=$ & $O((\log x)^{\frac{y}{\log y}})$\\
\end{tabular}\end{center}
\begin{exercise}
Find a decent upper bound for $\psi(x,y)$.
\end{exercise}
\subsection{Lower Bound on the Density}
The highest power of $y$ that can divide into $x$ is $\log_y(x)$. It is easily seen that, $\sum_{i=1}^{t} \alpha_i \geq \log_y(x) + t$. Therefore,\begin{center}\begin{tabular}{lcl}
$\psi(x,y)$ & $\geq$ & $\nchoosek{\log_y(x) + t}{t}$ \\
 & $\geq$ & $\frac{t^{\log_y x}}{(\log_y x)!}$\\
 & $=$ & $\Omega \left(\begin{array}{c} \frac{t^{\log_y x}}{(\log_y x)^{\log_y x}} \end{array}\right)$\\
 & $=$ & $\Omega \left(\begin{array}{c} \frac{x}{(\log_y x)^{2\log_y x}} \end{array}\right)$\\
\end{tabular}\end{center}

\section{References}
\begin{enumerate}
\item Eric W. Weisstein. "Smooth Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SmoothNumber.html
\end{enumerate}
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