introduction to number theory

In order to navigate out of this carousel please use your heading shortcut key to navigate to the next or previous heading. \end{equation*}. }\) If $k = -2\text{,}$ we get $(57, -32)\text{.}$. Here is a motivating example. Watch the recordings here on Youtube! $ \def\dbland{\bigwedge \!\!\bigwedge}$ Your recently viewed items and featured recommendations, Select the department you want to search in. }\) So $kn$ must also be divisible by $d\text{. In fact, we really only need to check one divisor of \(a$ and $n\text{:}$ the greatest common divisor. $ \def\rng{\mbox{range}}$ Equations which are intended to only have integer solutions were first studied by in the third century by the Greek mathematician Diophantus of Alexandria, and as such are called Diophantine equations. If $k = 0\text{,}$ the solution is $(-1,2)\text{,}$ and yes, $-17 + 2\cdot 29 = 41\text{. This means that \(x = 9$ and $x = 14$ and $x = 19$ and so on will each also be a solution because as we saw above, replacing any number in a congruence with a congruent number does not change the truth of the congruence. Notice we also include negative integers. Describe the remainder classes modulo $5\text{.}$. An equation in two or more variables is called a Diophantine equation if only integers solutions are of interest. }\) Thus, \begin{equation*} \begin{aligned}84x - 38 \amp \equiv 79 \pmod{15} \\ 9x - 8 \amp \equiv 4 \pmod{15} \\ 9x \amp \equiv 12 \pmod{15} \\ 9x \amp \equiv 72 \pmod{15}. A more common technique would be to apply the, \begin{equation*} 5x + 8y = 637. While we don't really know how to divide, we do know how to multiply. ―IACR Book Reviews, May 2011, Erickson and Vazzana provide a solid book, comprising 12 chapters, for courses in this area … All in all, a welcome addition to the stable of elementary number theory works for all good undergraduate libraries.―J. This will allow us to reduce the congruence to just one variable. For any integers $a\text{,}$ $b\text{,}$ and $n\text{,}$ we have, \begin{equation*} a \equiv b \pmod{n} \qquad \mbox{ if and only if } \qquad n \mid a-b. 1, August 2009, … reader-friendly text … 'Highlighting both fundamental and advanced topics, this introduction provides all of the tools to achieve a solid foundation in number theory. $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$, [ "article:topic", "license:ccbysa", "showtoc:no", "authorname:olevin", "remainder class modulo" ], $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$, (Template:MathJaxLevin), /content/body/div/p[1]/span, line 1, column 11, (Bookshelves/Combinatorics_and_Discrete_Mathematics/Book:_Discrete_Mathematics_(Levin)/5:_Additional_Topics/5.2:_Introduction_to_Number_Theory), /content/body/p/span, line 1, column 22, True. Article by Vicky Neale. $ \def\circleA{(-.5,0) circle (1)}$ This is the main question of number theory: a huge, ancient, complex, and above all, beautiful branch of mathematics. Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Think of congruence as being “basically equal.” If we have two numbers which are basically equal, and we add basically the same thing to both sides, the result will be basically equal. True. If $a \equiv b \pmod{n}$ then $b \equiv a \pmod{n}\text{. 2 Again, there is a mathematical theory of equivalence relations which applies in many more instances than the one we look at here. That is, they are the same up to division by \(b$. Topics covered include distribution of primes, unique factorization, reduction of positive definite quadratic forms, the Kronecker symbol, continued fractions, and what Gauss did. }\) If $k = 3\text{,}$ the solution is $(-88, 53)\text{. }$ This is because 0 is a multiple of every number: $0 = x\cdot 0\text{. \( \def\E{\mathbb E}$ It is either true or false. We will answer these questions, and more, after first investigating some simpler properties of numbers themselves. }\) From the division algorithm, we know there will be exactly 5 remainder classes, because there are only 5 choices for what $r$ could be ($0 \le r < 5$). However. $ \def\ansfilename{practice-answers}$ $ \def\VVee{\d\Vee\mkern-18mu\Vee}$ Would there be some amount after which all amounts would be possible? '―L’Enseignement Mathématique, Vol. 46, No. }\), This says that one way to make $6.37 is to take 121 of the 5-cent stamps and 4 of the 8-cent stamps. Please try again. This has changed in recent years however, as applications of number theory have been unearthed. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. \draw (\x,\y) +(90:\r) -- +(30:\r) -- +(-30:\r) -- +(-90:\r) -- +(-150:\r) -- +(150:\r) -- cycle; 1 It is possible to develop a mathematical theory of partitions, prove statements about all partitions in general and then apply those observations to our case here. $ \def\circleClabel{(.5,-2) node[right]{$C$}}$ 4 “goes into” 20 five times without remainder. This was the right set of numbers to work with in discrete mathematics because we always dealt with a whole number of things. To get the free app, enter your mobile phone number. We will work in numbers of cents. You're listening to a sample of the Audible audio edition. For each $k\text{,}$ $x = -1-29k$ and $y = 2 + 17k$ will satisfy the equation. Take the largest factor of both $d$ and $n\text{,}$ and cancel that out from $n\text{. \( \def\var{\mbox{var}}$ $ \def\circleC{(0,-1) circle (1)}$ True. Let's also see how you could solve this using our rules for the algebra of congruences. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. First, we need a Diophantine equation. Negatives? Thus we can simplify further: \begin{equation*} 6^{41} = 6\cdot (6^2)^{20} \equiv 6 \cdot 1^{20} \pmod 7. In our case, we reduce the congruence as follows: Now at this point we know $y = 2 + 17k$ will work for any integer $k\text{. Again, we are appealing to our claim that we can replace congruent elements, but we are really appealing to property 3 about the arithmetic of congruence: we know \(100 \equiv 1 \pmod{9}\text{,}$ so if we multiply both sides by $4\text{,}$ we get $400 \equiv 4 \pmod 9\text{. We choose 17 because \(17x$ will have remainder 0. What sorts of questions belong to the realm of number theory? First, check if perhaps there are no solutions because a divisor of $51$ and $87$ is not a divisor of $123\text{. \( \def\circleAlabel{(-1.5,.6) node[above]{$A$}}$ After viewing product detail pages, look here to find an easy way to navigate back to pages you are interested in. If we subtract, we get $136299 - 136286 = 13\text{. 54, No. The other two facts can be proved in a similar way. In fact, \(x \mid 0$ is true for all $x\text{. \( \def\threesetbox{(-2.5,-2.4) rectangle (2.5,1.4)}$ Chapman and Hall/CRC; 1st edition (October 30, 2007), Reviewed in the United States on August 3, 2008. Note that in the example above, every integer is in exactly one remainder class. $ \def\con{\mbox{Con}}$ \end{aligned} \end{equation*}. We work hard to protect your security and privacy.