From 688ef309c4bcfbea41e84ee5f842d16080dc815f Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Sat, 25 Feb 2017 22:56:49 +0200 Subject: [PATCH] References etc. --- chapter21.tex | 24 ++++++++++++++---------- list.tex | 10 ++++++++++ 2 files changed, 24 insertions(+), 10 deletions(-) diff --git a/chapter21.tex b/chapter21.tex index ebd31d4..c49f96f 100644 --- a/chapter21.tex +++ b/chapter21.tex @@ -24,7 +24,7 @@ z = \sqrt[3]{3}.\\ However, nobody knows if there are any three \emph{integers} $x$, $y$ and $z$ 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 and algorithms in number theory. @@ -205,7 +205,7 @@ so the result of the function is $[2,2,2,3]$. \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 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. @@ -327,7 +327,8 @@ The greatest common divisor and the least common multiple are connected as follows: \[\textrm{lcm}(a,b)=\frac{ab}{\textrm{gcd}(a,b)}\] -\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. The algorithm is based on the following formula: \begin{equation*} @@ -354,7 +355,8 @@ For example, Numbers $a$ and $b$ are \key{coprime} 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 $n$ between $1$ and $n$. For example, $\varphi(12)=4$, @@ -432,12 +434,12 @@ int modpow(int x, int n, int m) { \index{Fermat'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 \[x^{m-1} \bmod m = 1\] when $m$ is prime and $x$ and $m$ are coprime. This also yields \[x^k \bmod m = x^{k \bmod (m-1)} \bmod m.\] -More generally, \key{Euler's theorem} states that +More generally, \key{Euler's theorem}\footnote{Euler published this theorem in 1763.} states that \[x^{\varphi(m)} \bmod m = 1\] when $x$ and $m$ are coprime. Fermat's theorem follows from Euler's theorem, @@ -517,7 +519,8 @@ cout << x*x << "\n"; // 2537071545 \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 \[ ax + by = c, \] where $a$, $b$ and $c$ are constants and we should find the values of $x$ and $y$. @@ -637,7 +640,7 @@ are solutions. \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., $a^2+b^2+c^2+d^2$. 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{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 as a sum of Fibonacci numbers such that no two numbers are equal or consecutive @@ -689,7 +693,7 @@ produces the smallest Pythagorean triple \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 \[(n-1)! \bmod n = n-1.\] For example, the number 11 is prime, because diff --git a/list.tex b/list.tex index 2a4082b..e0d0669 100644 --- a/list.tex +++ b/list.tex @@ -25,6 +25,11 @@ On a routing problem. \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} M. A. Bender and M. Farach-Colton. The LCA problem revisited. In @@ -301,4 +306,9 @@ \emph{Des Rösselsprunges einfachste und allgemeinste Lösung}. 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} \ No newline at end of file