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 
+     LÃ¥t \(\tau(n)\) 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
+    LÃ¥t \(p\) 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
+   LÃ¥t \(\tau(n)\) 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
+  LÃ¥t \(p\) 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
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+++ b/TATA54/TENTOR/.#tentaTATA5420240601sketch.org
@@ -0,0 +1 @@
+jansn@jansn19-thinkpad.5341:1714983856
\ No newline at end of file
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new file mode 100644
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diff --git a/TATA54/TENTOR/tenta-TATA54-20240601.tex b/TATA54/TENTOR/tenta-TATA54-20240601.tex
new file mode 100644
index 0000000..775f110
--- /dev/null
+++ b/TATA54/TENTOR/tenta-TATA54-20240601.tex
@@ -0,0 +1,98 @@
+\documentclass[12pt,a4paper]{article}
+%\usepackage{times}
+%\usepackage{euler}
+\usepackage{a4wide}
+\usepackage{enumerate}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage[utf8]{inputenc}
+\usepackage[swedish]{babel}
+\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 
+     LÃ¥t \(\tau(n)\) 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
+    LÃ¥t \(p\) 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 
+     LÃ¥t \(\tau(n)\) 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
+    LÃ¥t \(p\) 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
+
+! Emergency stop.
+<*> #tentaTATA5420240601.tex#.tex
+                                 
+*** (job aborted, file error in nonstop mode)
+
+ 
+Here is how much of TeX's memory you used:
+ 3 strings out of 476041
+ 121 string characters out of 5793174
+ 1849388 words of memory out of 6000000
+ 20559 multiletter control sequences out of 15000+600000
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+ 1137 hyphenation exceptions out of 8191
+ 0i,0n,0p,1b,6s stack positions out of 10000i,1000n,20000p,200000b,200000s
+!  ==> Fatal error occurred, no output PDF file produced!
-- 
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