diff --git a/TATA54/TENTOR/#tentaTATA5420240601sketch.org# b/TATA54/TENTOR/#tentaTATA5420240601sketch.org#
index b6a290e2bfd7f4b574e22fabb5b5a84427135172..6a3c47b0e9970d0e3e4e32b45376a9600fcc619c 100644
--- a/TATA54/TENTOR/#tentaTATA5420240601sketch.org#
+++ b/TATA54/TENTOR/#tentaTATA5420240601sketch.org#
@@ -47,11 +47,15 @@ def H_L_tree(f, p: int, r: int):
q = p^k
R.<x> = ZZ[]
f = x^5 + 2*x + 3
+ df=f.diff(); df
+ f.roots(Integers(p),multiplicities=False)
f.roots(Integers(q),multiplicities=False)
-
+
#+end_src
#+RESULTS:
+: 5*x^4 + 2
+: [0, 2, 1]
: [12, 26, 22]
#+begin_src sage :session :results file :file "u1.png"
diff --git a/TATA54/TENTOR/tenta-TATA54-20240601-losning.aux b/TATA54/TENTOR/tenta-TATA54-20240601-losning.aux
new file mode 100644
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+++ b/TATA54/TENTOR/tenta-TATA54-20240601-losning.aux
@@ -0,0 +1,6 @@
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+\providecommand\babel@aux[2]{}
+\@nameuse{bbl@beforestart}
+\catcode `"\active
+\babel@aux{swedish}{}
+\gdef \@abspage@last{2}
diff --git a/TATA54/TENTOR/tenta-TATA54-20240601-losning.log b/TATA54/TENTOR/tenta-TATA54-20240601-losning.log
new file mode 100644
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diff --git a/TATA54/TENTOR/tenta-TATA54-20240601-losning.pdf b/TATA54/TENTOR/tenta-TATA54-20240601-losning.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..4e982f03e0d8585d5b7206b310d920d863515aab
Binary files /dev/null and b/TATA54/TENTOR/tenta-TATA54-20240601-losning.pdf differ
diff --git a/TATA54/TENTOR/tenta-TATA54-20240601-losning.tex b/TATA54/TENTOR/tenta-TATA54-20240601-losning.tex
new file mode 100644
index 0000000000000000000000000000000000000000..976b17052c7c7d014a245ddac7d95fd38df46988
--- /dev/null
+++ b/TATA54/TENTOR/tenta-TATA54-20240601-losning.tex
@@ -0,0 +1,130 @@
+\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{\Reals}{\boldsymbol{R}}
+\newcommand{\QQ}{\boldsymbol{QQ}}
+\newcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}
+\newcommand{\divides}[2]{{#1} \lvert {#2}}
+\newcommand{\soln}{\textbf Lösning: }
+\pagestyle{empty}
+\begin{document}
+\noindent{
+Talteori 6hp, Kurskod TATA54, Provkod TEN1 \\
+1 juni 2024 \\
+LINKÖPINGS UNIVERSITET\\
+Matematiska Institutionen\\
+Examinator: Jan Snellman}
+
+\bigskip
+Lösningsförslag
+
+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.
+ \soln
+ Polynomet \(f(x) = x^5 + 2x +3 \equiv x(x^4-1) \mod 3\),
+ så det har nollställena 0,1,-1 modulo 3.
+ Den formella derivatan är
+ \(f'(x)=5x^4+2 \equiv -(x^4+1) \mod 3\) vilket är
+ skiljt från noll evaluerat på de tre nollställena.
+ Alla dessa tre nollställen lyfter alltså unikt till
+ nollställen modulo \(3^k\) enligt Hensels lemma,
+ så det finns 3 nollställen modulo 27 och modulo 81.
+ Modulo 27 är dessa 12, 22, och 26.
+
+
+\item Hitta alla rationella punkter på kurvan
+ \begin{displaymath}
+ x^2 -7y^2 - 1 = 0
+ \end{displaymath}
+
+ \soln Parametrisera rationellt genom
+ att skicka \(k \in \Reals\)
+ till den unika skärningspunkten mellan kurvan och linjen
+ med lutning \(k\) genom \((-1,0)\). Det ger en bijektion
+ mellan kurvans högra gren och ett intervall av reella linjen;
+ denna bijektion skickar rationella \(k\) till
+ rationella punkter, och omvänt. Använd symmetri för att
+ få med vänstra grenen.
+
+
+\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.
+
+ \soln
+ En punkt \((x,y)\) på kurvan uppfyller antingen
+ \(x^2-7y^2=1\) eller
+ \(x^2-7y^2=-1\). Heltalspunkter är således antingen lösningar
+ till Pells ekvation, eller till ``negativa Pell''.
+ Kedjebråksutvecklingen är
+ \[\sqrt{7} = [2; \overline{1,1,1,4}] \]
+ Sats i Rosen ger att
+\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-losning.tex~ b/TATA54/TENTOR/tenta-TATA54-20240601-losning.tex~
new file mode 100644
index 0000000000000000000000000000000000000000..775f11077a2530537d4f0d5677bd747e18c36d21
--- /dev/null
+++ b/TATA54/TENTOR/tenta-TATA54-20240601-losning.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:
+
+