From f3e8a75a807b4a7da8b5fb7398f6117b296e67a6 Mon Sep 17 00:00:00 2001 From: Jan Snellman <jan.snellman@liu.se> Date: Mon, 20 May 2024 08:44:46 +0200 Subject: [PATCH] tenta 1 juni --- TATA54/TENTOR/#tentaTATA5420240601.tex# | 79 ++++--- TATA54/TENTOR/#tentaTATA5420240601sketch.org# | 161 +++++++++++++ TATA54/TENTOR/.#tentaTATA5420240601sketch.org | 1 + TATA54/TENTOR/tenta-TATA54-20240601.aux | 6 + TATA54/TENTOR/tenta-TATA54-20240601.log | 213 ++++++++++++++++++ TATA54/TENTOR/tenta-TATA54-20240601.pdf | Bin 0 -> 118013 bytes TATA54/TENTOR/tenta-TATA54-20240601.tex | 98 ++++++++ TATA54/TENTOR/tenta-TATA54-20240601.tex~ | 97 ++++++++ TATA54/TENTOR/texput.log | 22 ++ 9 files changed, 642 insertions(+), 35 deletions(-) create mode 100644 TATA54/TENTOR/#tentaTATA5420240601sketch.org# create mode 120000 TATA54/TENTOR/.#tentaTATA5420240601sketch.org create mode 100644 TATA54/TENTOR/tenta-TATA54-20240601.aux create mode 100644 TATA54/TENTOR/tenta-TATA54-20240601.log create mode 100644 TATA54/TENTOR/tenta-TATA54-20240601.pdf create mode 100644 TATA54/TENTOR/tenta-TATA54-20240601.tex create mode 100644 TATA54/TENTOR/tenta-TATA54-20240601.tex~ create mode 100644 TATA54/TENTOR/texput.log diff --git a/TATA54/TENTOR/#tentaTATA5420240601.tex# b/TATA54/TENTOR/#tentaTATA5420240601.tex# index 691177b..f0ef930 100644 --- a/TATA54/TENTOR/#tentaTATA5420240601.tex# +++ b/TATA54/TENTOR/#tentaTATA5420240601.tex# @@ -38,41 +38,50 @@ som räknar antalet multiplikativt inverterbara kongruensklasser modulo \(n\).}} \begin{enumerate}[1)] - \item Hitta alla lösningar till - \begin{displaymath} - x^5 +2x +3 \equiv 0 \mod 81 - \end{displaymath} - - - - \item Bestäm en primitiv rot modulo 7, och hitta sedan - alla heltalslösningar till - \begin{displaymath} - x^x \equiv x \mod 7 - \end{displaymath} - (Obs: \(19^{19} \equiv 19 \mod 7\) och \(19 \equiv 5 \mod 7\), - men \(5^5 \equiv 3 \mod 7\).) - \item Bevisa att - \begin{math} - \sum_{\divides{d}{n}} \phi(d)=n - \end{math} - och med hjälp av detta att - \begin{math} - \sum_{\divides{d}{n}} d\mu(n/d)=\phi(n). - \end{math} - (Du behöver inte visa att \(\phi\) och \(\mu\) är multiplikativa). - \item Ange alla udda primtal \(p\) så att kongruensen \(x^2 \equiv 11 \mod p\) - är lösbar. - - \item Hitta den lösning \((x,y)\) i positiva heltal till - \begin{displaymath} - x^2-11y^2=1 - \end{displaymath} - som ligger närmast origo. Hitta sedan en annan lösning i positiva heltal. - Ge sedan en rationell approximation till \(\sqrt{11}\) med absolutfel \(<10^{-4}\). - - \item Skriv 87 som en summa av fyra kvadrater. - Visa sedan att det inte går med tre. +\item Hitta alla lösningar till + \begin{displaymath} + x^5 +2x +3 \equiv 0 \mod 27 + \end{displaymath} + och bestäm \textbf{antalet} sådana lösningar modulo 81. + +\item Hitta alla rationella punkter på kurvan + \begin{displaymath} + x^2 -7y^2 - 1 = 0 + \end{displaymath} + +\item + Ange explicit två olika punkter på kurvan + \begin{displaymath} + (x^2 -7y^2 - 1)(x^2-7y^2+1)=0 + \end{displaymath} + vars koordinater är positiva heltal; beskriv alla övriga + sådana punkter. + +\item + Låt \(\tau(n)\) beteckna antalet positiva delare till heltalet \(n\). + Visa att + \[ + \sum_{\divides{k}{n}} \tau(k) + = \prod_{j=1}^r \binom{a_j + 2}{2} + \] + då \(n\) har primfaktoriseringen \(n=\prod_{j=1}^r p_j^{a_j}\). + Visa även att + \[ + \sum_{k=1}^n \tau(k) = + \sum_{k=1}^n \left \lfloor \frac{n}{k} \right \rfloor + \] + + \item + Låt \(p\) vara ett udda primtal. Visa att ekvationen + \[ x^4 \equiv -1 \mod p\] + är lösbar om och endast om \(p \equiv 1 \mod 8\). + +\item Vad är den största multiplikativa ordningen + av ett heltal modulo 99? + Ange något sådant heltal av maximal multiplikativ ordning. + +\item Visa att \(n^3(n^{18}-1) \equiv 0 \mod 26\) + för alla heltal \(n\). \end{enumerate} diff --git a/TATA54/TENTOR/#tentaTATA5420240601sketch.org# b/TATA54/TENTOR/#tentaTATA5420240601sketch.org# new file mode 100644 index 0000000..b6a290e --- /dev/null +++ b/TATA54/TENTOR/#tentaTATA5420240601sketch.org# @@ -0,0 +1,161 @@ +#+title: TATA54 tenta 20240601, lösningskiss +#+author: Jan Snellman + +* Preliminärt +7 uppgifter, 3p för varje. Försöker täcka +- Kongruensräkning +- Hensellyft +- Kvadratisk reciprocite +- Kedjebråk +- Pell eller Pytagoranska tripplar + + +* Gemensamm kod +#+begin_src sage :session :export none +def H_L_tree(f, p: int, r: int): + vert = [(0,0)] + for j in range(1,r+1): + fj = f.change_ring(Integers(p^j)) + fzj = fj.roots(multiplicities=False) + vert += [(z,j) for z in fzj] + return DiGraph([vert, + lambda u,v: (u[1] == 1 and v[1] == 0) + or + ( (u[1] == v[1]+1) and ((u[0] - v[0]) % p^v[1] ==0) ) + ]) +#+end_src + +#+RESULTS: + +#+begin_src latex + \newcommand{\QQ}{\mathbf{Q}} + \newcommand{\ZZ}{\mathbf{Z}} +#+end_src +* Uppgifterna +** U1 +#+begin_src latex + Hitta alla lösningar till + \begin{displaymath} + x^5 +2x +3 \equiv 0 \mod 27 + \end{displaymath} +#+end_src + + +#+begin_src sage :session + p = 3 + k = 3 + q = p^k + R.<x> = ZZ[] + f = x^5 + 2*x + 3 + f.roots(Integers(q),multiplicities=False) + +#+end_src + +#+RESULTS: +: [12, 26, 22] + +#+begin_src sage :session :results file :file "u1.png" + H_L_tree(f,p,k).show(layout='tree') +#+end_src + +#+RESULTS: +[[file:u1.png]] + + +** U2 + + +#+begin_src latex + Hitta alla rationella punkter på kurvan + \begin{displaymath} + x^2 -7y^2 - 1 = 0 + \end{displaymath} +#+end_src + +Parametrisera med lutning, rationell punkt (-1,0) + + +** U3 + +#+begin_src latex + Ange explicit två punkter på kurvan + \begin{displaymath} + (x^2 -7y^2 - 1)(x^2-7y^2+1)=0 + \end{displaymath} + vars koordinater är positiva heltal; beskriv alla övriga sådana. +#+end_src + +Antingen lösning till Pell x^2-7y^2=1 eller till negativa Pell x^2-7y^2=-1. + +#+begin_src sage :session + d = 7 + cfd = continued_fraction(sqrt(d)) + cfd + for j in range(15): + co = cfd.convergent(j) + y = co.denominator() + x = co.numerator() + print(j,co,x,y,x^2-d*y^2) +#+end_src + +#+RESULTS: +#+begin_example +[2; 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, ...] +0 2 2 1 -3 +1 3 3 1 2 +2 5/2 5 2 -3 +3 8/3 8 3 1 +4 37/14 37 14 -3 +5 45/17 45 17 2 +6 82/31 82 31 -3 +7 127/48 127 48 1 +8 590/223 590 223 -3 +9 717/271 717 271 2 +10 1307/494 1307 494 -3 +11 2024/765 2024 765 1 +12 9403/3554 9403 3554 -3 +13 11427/4319 11427 4319 2 +14 20830/7873 20830 7873 -3 +#+end_example + + +** U4 +#+begin_src latex + Låt \(\tau(n)\) beteckna antalet positiva delare till heltalet \(n\). + Visa att + \[ + \sum_{\divides{k}{n}} \tau(k) + = \prod_{j=1}^r \binom{a_j + 2}{2} + \] + då \(n\) har primfaktoriseringen \(n=\prod_{j=1}^r p_j^{a_j}\). + Visa även att + \[ + \sum_{k=1}^n \tau(k) = \sum_{k=1}^n \left \lfloor \frac{n}{k} \right \rfloor + \] +#+end_src + + +** U5 +#+begin_src latex + Låt \(p\) vara ett udda primtal. Visa att ekvationen + \[ x^4 \equiv -1 \mod p\] + är lösbar om och endast om \(p \equiv 1 \mod 8\). +#+end_src + +** U6 +#+begin_src latex + Vad är den största multiplikativa ordningen av ett heltal modulo 99? + Ange något sådant heltal av maximal multiplikativ ordning. +#+end_src + +#+RESULTS: +#+begin_export latex +Vad är den största multiplikativa ordningen av ett heltal modulo 99? +Ange något sådant heltal av maximal multiplikativ ordning. +#+end_export + +** U7 +#+begin_src latex + Visa att \(n^3(n^{18}-1) \equiv 0 \mod 26\) för alla heltal \(n\). +#+end_src +* Slut diff --git a/TATA54/TENTOR/.#tentaTATA5420240601sketch.org b/TATA54/TENTOR/.#tentaTATA5420240601sketch.org new file mode 120000 index 0000000..091951a --- /dev/null +++ b/TATA54/TENTOR/.#tentaTATA5420240601sketch.org @@ -0,0 +1 @@ +jansn@jansn19-thinkpad.5341:1714983856 \ No newline at end of file diff --git a/TATA54/TENTOR/tenta-TATA54-20240601.aux b/TATA54/TENTOR/tenta-TATA54-20240601.aux new file mode 100644 index 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+\begin{document} +\noindent{ +Talteori 6hp, Kurskod TATA54, Provkod TEN1 \\ +1 juni 2024 \\ +LINKÖPINGS UNIVERSITET\\ +Matematiska Institutionen\\ +Examinator: Jan Snellman} + +\bigskip + + +Alla problem ger maximalt 3 poäng. Full poäng kräver fullständig lösning. +8p räcker för betyg 3, 11p för betyg 4, 14p för betyg 5. + +\medskip + +{\small{Med \(\mu\) avses Möbiusfunktionen, den multiplikativa funktion som uppfyller +\(\mu(1)=1\), \(\mu(p)=-1\) för \(p\) primtal, och \(\mu(n)=0\) för alla \(n>1\) som +delas av kvadraten av något primtal. Med \(\phi\) avses Eulers phi-funktion, +som räknar antalet multiplikativt inverterbara kongruensklasser modulo \(n\).}} + +\medskip + +\begin{enumerate}[1)] + +\item Hitta alla lösningar till + \begin{displaymath} + x^5 +2x +3 \equiv 0 \mod 27 + \end{displaymath} + och bestäm \textbf{antalet} sådana lösningar modulo 81. + +\item Hitta alla rationella punkter på kurvan + \begin{displaymath} + x^2 -7y^2 - 1 = 0 + \end{displaymath} + +\item + Ange explicit två olika punkter på kurvan + \begin{displaymath} + (x^2 -7y^2 - 1)(x^2-7y^2+1)=0 + \end{displaymath} + vars koordinater är positiva heltal; beskriv alla övriga + sådana punkter. + +\item + Låt \(\tau(n)\) beteckna antalet positiva delare till heltalet \(n\). + Visa att + \[ + \sum_{\divides{k}{n}} \tau(k) + = \prod_{j=1}^r \binom{a_j + 2}{2} + \] + då \(n\) har primfaktoriseringen \(n=\prod_{j=1}^r p_j^{a_j}\). + Visa även att + \[ + \sum_{k=1}^n \tau(k) = + \sum_{k=1}^n \left \lfloor \frac{n}{k} \right \rfloor + \] + + \item + Låt \(p\) vara ett udda primtal. Visa att ekvationen + \[ x^4 \equiv -1 \mod p\] + är lösbar om och endast om \(p \equiv 1 \mod 8\). + +\item Vad är den största multiplikativa ordningen + av ett heltal modulo 99? + Ange något sådant heltal av maximal multiplikativ ordning. + +\item Visa att \(n^3(n^{18}-1) \equiv 0 \mod 26\) + för alla heltal \(n\). +\end{enumerate} + + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: + + diff --git a/TATA54/TENTOR/tenta-TATA54-20240601.tex~ b/TATA54/TENTOR/tenta-TATA54-20240601.tex~ new file mode 100644 index 0000000..f0ef930 --- /dev/null +++ b/TATA54/TENTOR/tenta-TATA54-20240601.tex~ @@ -0,0 +1,97 @@ +\documentclass[12pt,a4paper]{article} +%\usepackage{times} +%\usepackage{euler} +\usepackage{a4wide} +\usepackage{enumerate} +\usepackage{amsmath,amssymb,amsthm} +\usepackage[latin1]{inputenc} +\newcommand{\sgd}{\mathrm{sgd}} +\newcommand{\set}[1]{\left\{{#1}\right\}} +\newcommand{\vek}[1]{\boldsymbol{#1}} +\newcommand\setsuchas[2]{\left\{\,{#1}\,\vrule\,{#2}\,\right\}} +\newcommand{\Z}{\boldsymbol{Z}} +\newcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}} +\newcommand{\divides}[2]{{#1} \lvert {#2}} +\pagestyle{empty} +\begin{document} +\noindent{ +Talteori 6hp, Kurskod TATA54, Provkod TEN1 \\ +1 juni 2024 \\ +LINKÖPINGS UNIVERSITET\\ +Matematiska Institutionen\\ +Examinator: Jan Snellman} + +\bigskip + + +Alla problem ger maximalt 3 poäng. Full poäng kräver fullständig lösning. +8p räcker för betyg 3, 11p för betyg 4, 14p för betyg 5. + +\medskip + +{\small{Med \(\mu\) avses Möbiusfunktionen, den multiplikativa funktion som uppfyller +\(\mu(1)=1\), \(\mu(p)=-1\) för \(p\) primtal, och \(\mu(n)=0\) för alla \(n>1\) som +delas av kvadraten av något primtal. Med \(\phi\) avses Eulers phi-funktion, +som räknar antalet multiplikativt inverterbara kongruensklasser modulo \(n\).}} + +\medskip + +\begin{enumerate}[1)] + +\item Hitta alla lösningar till + \begin{displaymath} + x^5 +2x +3 \equiv 0 \mod 27 + \end{displaymath} + och bestäm \textbf{antalet} sådana lösningar modulo 81. + +\item Hitta alla rationella punkter på kurvan + \begin{displaymath} + x^2 -7y^2 - 1 = 0 + \end{displaymath} + +\item + Ange explicit två olika punkter på kurvan + \begin{displaymath} + (x^2 -7y^2 - 1)(x^2-7y^2+1)=0 + \end{displaymath} + vars koordinater är positiva heltal; beskriv alla övriga + sådana punkter. + +\item + Låt \(\tau(n)\) beteckna antalet positiva delare till heltalet \(n\). + Visa att + \[ + \sum_{\divides{k}{n}} \tau(k) + = \prod_{j=1}^r \binom{a_j + 2}{2} + \] + då \(n\) har primfaktoriseringen \(n=\prod_{j=1}^r p_j^{a_j}\). + Visa även att + \[ + \sum_{k=1}^n \tau(k) = + \sum_{k=1}^n \left \lfloor \frac{n}{k} \right \rfloor + \] + + \item + Låt \(p\) vara ett udda primtal. Visa att ekvationen + \[ x^4 \equiv -1 \mod p\] + är lösbar om och endast om \(p \equiv 1 \mod 8\). + +\item Vad är den största multiplikativa ordningen + av ett heltal modulo 99? + Ange något sådant heltal av maximal multiplikativ ordning. + +\item Visa att \(n^3(n^{18}-1) \equiv 0 \mod 26\) + för alla heltal \(n\). +\end{enumerate} + + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: + + diff --git a/TATA54/TENTOR/texput.log b/TATA54/TENTOR/texput.log new file mode 100644 index 0000000..ea18a22 --- /dev/null +++ b/TATA54/TENTOR/texput.log @@ -0,0 +1,22 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023) (preloaded format=pdflatex 2024.5.13) 20 MAY 2024 08:40 +entering extended mode + restricted \write18 enabled. + file:line:error style messages enabled. + %&-line parsing enabled. +**#tentaTATA5420240601.tex#.tex + +! 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