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Antti H S Laaksonen 2017-02-25 22:56:49 +02:00
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24
chapter21.tex
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@ -24,7 +24,7 @@ z = \sqrt[3]{3}.\\
However, nobody knows if there are any three However, nobody knows if there are any three
\emph{integers} $x$, $y$ and $z$ \emph{integers} $x$, $y$ and $z$
that would satisfy the equation, but this that would satisfy the equation, but this
is an open problem in number theory. is an open problem in number theory \cite{bec07}.
In this chapter, we will focus on basic concepts In this chapter, we will focus on basic concepts
and algorithms in number theory. and algorithms in number theory.
@ -205,7 +205,7 @@ so the result of the function is $[2,2,2,3]$.
\index{sieve of Eratosthenes} \index{sieve of Eratosthenes}
The \key{sieve of Eratosthenes} is a preprocessing The \key{sieve of Eratosthenes}\footnote{Eratosthenes (c. 276 BC -- c. 194 BC) was a Greek mathematician.} is a preprocessing
algorithm that builds an array using which we algorithm that builds an array using which we
can efficiently check if a given number between $2 \ldots n$ can efficiently check if a given number between $2 \ldots n$
is prime and, if it is not, find one prime factor of the number. is prime and, if it is not, find one prime factor of the number.
@ -327,7 +327,8 @@ The greatest common divisor and the least common multiple
are connected as follows: are connected as follows:
\[\textrm{lcm}(a,b)=\frac{ab}{\textrm{gcd}(a,b)}\] \[\textrm{lcm}(a,b)=\frac{ab}{\textrm{gcd}(a,b)}\]
\key{Euclid's algorithm} provides an efficient way \key{Euclid's algorithm}\footnote{Euclid was a Greek mathematician who
lived in about 300 BC. This is perhaps the first known algorithm in history.} provides an efficient way
to find the greatest common divisor of two numbers. to find the greatest common divisor of two numbers.
The algorithm is based on the following formula: The algorithm is based on the following formula:
\begin{equation*} \begin{equation*}
@ -354,7 +355,8 @@ For example,
Numbers $a$ and $b$ are \key{coprime} Numbers $a$ and $b$ are \key{coprime}
if $\textrm{gcd}(a,b)=1$. if $\textrm{gcd}(a,b)=1$.
\key{Euler's totient function} $\varphi(n)$ \key{Euler's totient function} $\varphi(n)$\footnote{Euler
presented this function in 1763.}
gives the number of coprime numbers to $n$ gives the number of coprime numbers to $n$
between $1$ and $n$. between $1$ and $n$.
For example, $\varphi(12)=4$, For example, $\varphi(12)=4$,
@ -432,12 +434,12 @@ int modpow(int x, int n, int m) {
\index{Fermat's theorem} \index{Fermat's theorem}
\index{Euler's theorem} \index{Euler's theorem}
\key{Fermat's theorem} states that \key{Fermat's theorem}\footnote{Fermat discovered this theorem in 1640.} states that
\[x^{m-1} \bmod m = 1\] \[x^{m-1} \bmod m = 1\]
when $m$ is prime and $x$ and $m$ are coprime. when $m$ is prime and $x$ and $m$ are coprime.
This also yields This also yields
\[x^k \bmod m = x^{k \bmod (m-1)} \bmod m.\] \[x^k \bmod m = x^{k \bmod (m-1)} \bmod m.\]
More generally, \key{Euler's theorem} states that More generally, \key{Euler's theorem}\footnote{Euler published this theorem in 1763.} states that
\[x^{\varphi(m)} \bmod m = 1\] \[x^{\varphi(m)} \bmod m = 1\]
when $x$ and $m$ are coprime. when $x$ and $m$ are coprime.
Fermat's theorem follows from Euler's theorem, Fermat's theorem follows from Euler's theorem,
@ -517,7 +519,8 @@ cout << x*x << "\n"; // 2537071545
\index{Diophantine equation} \index{Diophantine equation}
A \key{Diophantine equation} is an equation of the form A \key{Diophantine equation}\footnote{Diophantus of Alexandria was a Greek
mathematician who lived in the 3th century.} is an equation of the form
\[ ax + by = c, \] \[ ax + by = c, \]
where $a$, $b$ and $c$ are constants where $a$, $b$ and $c$ are constants
and we should find the values of $x$ and $y$. and we should find the values of $x$ and $y$.
@ -637,7 +640,7 @@ are solutions.
\index{Lagrange's theorem} \index{Lagrange's theorem}
\key{Lagrange's theorem} states that every positive integer \key{Lagrange's theorem}\footnote{J.-L. Lagrange (1736--1813) was an Italian mathematician.} states that every positive integer
can be represented as a sum of four squares, i.e., can be represented as a sum of four squares, i.e.,
$a^2+b^2+c^2+d^2$. $a^2+b^2+c^2+d^2$.
For example, the number 123 can be represented For example, the number 123 can be represented
@ -648,7 +651,8 @@ as the sum $8^2+5^2+5^2+3^2$.
\index{Zeckendorf's theorem} \index{Zeckendorf's theorem}
\index{Fibonacci number} \index{Fibonacci number}
\key{Zeckendorf's theorem} states that every \key{Zeckendorf's theorem}\footnote{E. Zeckendorf published the theorem in 1972 \cite{zec72};
however, this was not a new result.} states that every
positive integer has a unique representation positive integer has a unique representation
as a sum of Fibonacci numbers such that as a sum of Fibonacci numbers such that
no two numbers are equal or consecutive no two numbers are equal or consecutive
@ -689,7 +693,7 @@ produces the smallest Pythagorean triple
\index{Wilson's theorem} \index{Wilson's theorem}
\key{Wilson's theorem} states that a number $n$ \key{Wilson's theorem}\footnote{J. Wilson (1741--1793) was an English mathematician.} states that a number $n$
is prime exactly when is prime exactly when
\[(n-1)! \bmod n = n-1.\] \[(n-1)! \bmod n = n-1.\]
For example, the number 11 is prime, because For example, the number 11 is prime, because

10
list.tex
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@ -25,6 +25,11 @@
On a routing problem. On a routing problem.
\emph{Quarterly of Applied Mathematics}, 16(1):87--90, 1958. \emph{Quarterly of Applied Mathematics}, 16(1):87--90, 1958.
\bibitem{bec07}
M. Beck, E. Pine, W. Tarrat and K. Y. Jensen.
New integer representations as the sum of three cubes.
\emph{Mathematics of Computation}, 76(259):1683--1690, 2007.
\bibitem{ben00} \bibitem{ben00}
M. A. Bender and M. Farach-Colton. M. A. Bender and M. Farach-Colton.
The LCA problem revisited. In The LCA problem revisited. In
@ -301,4 +306,9 @@
\emph{Des Rösselsprunges einfachste und allgemeinste Lösung}. \emph{Des Rösselsprunges einfachste und allgemeinste Lösung}.
Schmalkalden, 1823. Schmalkalden, 1823.
\bibitem{zec72}
E. Zeckendorf.
Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas.
\emph{Bull. Soc. Roy. Sci. Liege}, 41:179--182, 1972.
\end{thebibliography} \end{thebibliography}