From d6aeba4f9bd0ac5da58cc08d8404b745911d38d7 Mon Sep 17 00:00:00 2001
From: Jan Snellman <jan.snellman@liu.se>
Date: Wed, 22 May 2024 11:13:40 +0200
Subject: [PATCH] tenta 1 juni klar

---
 .../tenta-TATA54-20240601-losning.el          |  33 +++
 .../.auctex-auto/tenta-TATA54-20240601.el     |  28 +++
 .../TENTOR/tenta-TATA54-20240601-losning.aux  |   2 +-
 .../TENTOR/tenta-TATA54-20240601-losning.log  | 218 ++++++++---------
 .../TENTOR/tenta-TATA54-20240601-losning.pdf  | Bin 139190 -> 106733 bytes
 .../tenta-TATA54-20240601-losning.synctex.gz  | Bin 0 -> 16854 bytes
 .../TENTOR/tenta-TATA54-20240601-losning.tex  | 176 ++++++--------
 TATA54/TENTOR/tenta-TATA54-20240601.log       | 219 +++++++++---------
 TATA54/TENTOR/tenta-TATA54-20240601.pdf       | Bin 118013 -> 129852 bytes
 .../TENTOR/tenta-TATA54-20240601.synctex.gz   | Bin 0 -> 6995 bytes
 TATA54/TENTOR/tenta-TATA54-20240601.tex       |  46 ++--
 TATA54/TENTOR/tentaTATA5420240601sketch.org   | 107 ++++++++-
 TATA54/TENTOR/u2.png                          | Bin 0 -> 12457 bytes
 13 files changed, 492 insertions(+), 337 deletions(-)
 create mode 100644 TATA54/TENTOR/.auctex-auto/tenta-TATA54-20240601-losning.el
 create mode 100644 TATA54/TENTOR/.auctex-auto/tenta-TATA54-20240601.el
 create mode 100644 TATA54/TENTOR/tenta-TATA54-20240601-losning.synctex.gz
 create mode 100644 TATA54/TENTOR/tenta-TATA54-20240601.synctex.gz
 create mode 100644 TATA54/TENTOR/u2.png

diff --git a/TATA54/TENTOR/.auctex-auto/tenta-TATA54-20240601-losning.el b/TATA54/TENTOR/.auctex-auto/tenta-TATA54-20240601-losning.el
new file mode 100644
index 0000000..67409fe
--- /dev/null
+++ b/TATA54/TENTOR/.auctex-auto/tenta-TATA54-20240601-losning.el
@@ -0,0 +1,33 @@
+(TeX-add-style-hook
+ "tenta-TATA54-20240601-losning"
+ (lambda ()
+   (TeX-add-to-alist 'LaTeX-provided-class-options
+                     '(("article" "11pt" "a4paper")))
+   (TeX-add-to-alist 'LaTeX-provided-package-options
+                     '(("inputenc" "utf8") ("babel" "swedish")))
+   (TeX-run-style-hooks
+    "latex2e"
+    "article"
+    "art11"
+    "a4wide"
+    "enumerate"
+    "amsmath"
+    "amssymb"
+    "amsthm"
+    "inputenc"
+    "babel")
+   (TeX-add-symbols
+    '("mgm" 2)
+    '("divides" 2)
+    '("legendre" 2)
+    '("setsuchas" 2)
+    '("vek" 1)
+    '("set" 1)
+    "sgd"
+    "Z"
+    "Reals"
+    "QQ"
+    "ett"
+    "soln"))
+ :latex)
+
diff --git a/TATA54/TENTOR/.auctex-auto/tenta-TATA54-20240601.el b/TATA54/TENTOR/.auctex-auto/tenta-TATA54-20240601.el
new file mode 100644
index 0000000..a840c6c
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@@ -0,0 +1,28 @@
+(TeX-add-style-hook
+ "tenta-TATA54-20240601"
+ (lambda ()
+   (TeX-add-to-alist 'LaTeX-provided-class-options
+                     '(("article" "12pt" "a4paper")))
+   (TeX-add-to-alist 'LaTeX-provided-package-options
+                     '(("inputenc" "utf8") ("babel" "swedish")))
+   (TeX-run-style-hooks
+    "latex2e"
+    "article"
+    "art12"
+    "enumerate"
+    "amsmath"
+    "amssymb"
+    "amsthm"
+    "inputenc"
+    "babel")
+   (TeX-add-symbols
+    '("divides" 2)
+    '("legendre" 2)
+    '("setsuchas" 2)
+    '("vek" 1)
+    '("set" 1)
+    "sgd"
+    "Z"
+    "QQ"))
+ :latex)
+
diff --git a/TATA54/TENTOR/tenta-TATA54-20240601-losning.aux b/TATA54/TENTOR/tenta-TATA54-20240601-losning.aux
index 302e499..7aa2b6d 100644
--- a/TATA54/TENTOR/tenta-TATA54-20240601-losning.aux
+++ b/TATA54/TENTOR/tenta-TATA54-20240601-losning.aux
@@ -3,4 +3,4 @@
 \@nameuse{bbl@beforestart}
 \catcode `"\active 
 \babel@aux{swedish}{}
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+\gdef \@abspage@last{2}
diff --git a/TATA54/TENTOR/tenta-TATA54-20240601-losning.log b/TATA54/TENTOR/tenta-TATA54-20240601-losning.log
index a798e3a..0bafbd8 100644
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diff --git a/TATA54/TENTOR/tenta-TATA54-20240601-losning.tex b/TATA54/TENTOR/tenta-TATA54-20240601-losning.tex
index 114f1e6..ed0c3b1 100644
--- a/TATA54/TENTOR/tenta-TATA54-20240601-losning.tex
+++ b/TATA54/TENTOR/tenta-TATA54-20240601-losning.tex
@@ -1,4 +1,4 @@
-\documentclass[12pt,a4paper]{article}
+\documentclass[11pt,a4paper]{article}
 %\usepackage{times}
 %\usepackage{euler}
 \usepackage{a4wide}
@@ -21,7 +21,7 @@
 \pagestyle{empty}
 \begin{document}
 \noindent{
-{\Huge Lösningsförslag} \\
+{\Huge Lösningsförslag} \\ \\
 Talteori 6hp, Kurskod TATA54, Provkod TEN1 \\
 1 juni 2024 \\
 LINKÖPINGS UNIVERSITET\\
@@ -30,145 +30,123 @@ 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.
-
-  
-  \soln
-  Polynomet \(f(x) = x^5 + 2x +3 \equiv x(x^4-1) \mod 3\),
+\item
+  Sätt \(f(x) = x^5 + 2x +3\). Då är \(f(x) \equiv x(x^4-1) \mod 3\),
   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.
+        \(f'(x)=5x^4+2\), så \(f'(x) \equiv -(x^4+1) \mod 3\).
+        Alltså är \(f'(x)\) konstant noll evaluerat på \(\Z_{3}\).
+        Vidare: enligt Hensels lemma så lyfter dessa tre nollställen
+        unikt till
+  nollställen modulo \(3^k\), för \(k \ge 2\).
+  Det finns alltså 3 nollställen modulo 27 och modulo 81.
   Modulo 27 är dessa 12, 22, och 26.
 
+  \item
+  Eftersom \(99=3^2*11\) så ges denna maximala ordning
+  av \(\mgm(\phi(9),\phi(11)) = \mgm(6,10) = 30\),
+  enligt sats 9.22 i Rosen.
+  Vi ser att redan \(\mathrm{ord}_{99}(2)=30\).
   
-\item   Hitta alla rationella punkter på kurvan
-  \begin{displaymath}
-    x^2 -7y^2 - 1 = 0
-  \end{displaymath}
-
-  \soln Parametrisera rationellt genom
+\item Parametrisera rationellt genom
   att skicka \(t \in \Reals\)
   till den unika skärningspunkten mellan kurvan och linjen
-  med lutning \(t\) genom \((-1,0)\). Det ger en bijektion
-  mellan kurvans högra gren och ett intervall av reella linjen;
+        med lutning \(t\) genom \((-1,0)\).
+        Explicit så
+        \begin{displaymath}
+          \left(\frac{-1}{\sqrt{7}},\frac{1}{\sqrt{7}}\right) \ni t \mapsto \left(
+            \frac{1+7t^{2}}{1-7t^{2}}, \frac{2t}{1-7t^{2}}
+            \right)
+          \end{displaymath}
+        med invers
+        \begin{displaymath}
+          (x,y) \mapsto \frac{y}{x+1}.
+        \end{displaymath}
+        Det ger en bijektion
+        mellan kurvans högra gren och ett intervall av reella linjen;
   denna bijektion skickar rationella \(t\) 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
+    En punkt \((x,y)\) på kurvan uppfyller antingen
   \(x^2-7y^2=1\) eller
-  \(x^2-7y^2=-1\). Heltalspunkter är således antingen lösningar
+  \(x^2-7y^2=-1\). Positiva heltalspunkter är således antingen lösningar
   till Pells ekvation med \(d=7\), eller till ``negativa Pell''.
   Kedjebråksutvecklingen är
   \[\sqrt{7} = [2; \overline{1,1,1,4}] \]
-  Sats i Rosen ger att 
+  En sats i Rosen ger att
   om konvergenterna till ovanstående kedjebråksutveckling är
   \(P_k/Q_k\) så ges lösningarna till Pells ekvation
   av
   \[(x,y) = (Q_k, \, P_k) \qquad \text{ då } Q_k^2-7P_k =1\]
   och motsvarande för negativa Pell. Eftersom periodlängden 4
-  är jämn, så har negativa Pell inga lösningar, medan
+  är jämn, så säger en annan sats i Rosen att negativa Pell saknar lösningar, medan
   Pell har lösningarna ovan då \(k=3,7,11,15,\dots\).
   För \(k=3,7\) erhålls lösningarna \((8,3)\) och
-  \((127,48)\).
+        \((127,48)\).
+
+        Vi kan också få alla lösningar \((x_{k},y_{k})\) från den primitiva lösningen
+        \((x_{1},y_{1})=(8,3)\) via
+        \begin{displaymath}
+          (8 + 3\sqrt{7})^{k} = x_{k} + \sqrt{7}y_{k}·
+        \end{displaymath}
+        Speciellt så är
+        \begin{displaymath}
+          (8 + 3\sqrt{7})^{2} = 64 + 2*8*3*\sqrt{7} + 3^{2}*7 = 127 + 48\sqrt{7}.
+        \end{displaymath}
+
 
 \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
-  \[
-    \sum_{k=1}^n \tau(k) =
-    \sum_{k=1}^n  \left \lfloor \frac{n}{k} \right \rfloor
-  \]
-  
-  \soln Låt \(\ett(n)=1\) för alla positiva heltal \(n\).
-  Då är \(\tau = \ett * \ett\) och
+   Låt \(\ett(n)=1\) för alla positiva heltal \(n\).
+        Då är \(\tau(n) = \sum_{\divides{k}{n}}1\)
+        så \(\tau = \ett * \ett\) och
   \[h(n):= \sum_{\divides{k}{n}} \tau(k)
     = (\ett * \ett * \ett)(n)
   \]
   Eftersom \(\ett\) är multiplikativ, är även
   \(h=\ett * \ett * \ett\) det, så det räcker
   att verifiera likheten för fallet att \(n=p^s\) är
-  en primpotens. Vi har att
+        en primpotens. Delarna till \(p^{s}\) är \(p^{a}\) med \(0 \le a \le s\),
+        så vi har att
   \[
-    h(p^s) = \sum_{a+b+c=s} 1 = \binom{s+3-1}{3-1}
+        h(p^s) = \sum_{a+b+c=s} 1 = \binom{s+3-1}{3-1}
+        = \binom{s+2}{2}
   \]
   enligt urval med återläggning.
 
-  För den andra delen, sätt
-  \(A(n,k) = \lfloor \frac{n}{k} \rfloor\)
-  och notera att
-  \[
-    A(n,k) - A(n-1,k)
-    =
-    \begin{cases}
-      1 & \divides{k}{n} \\
-      0 & \text{ annars }
-    \end{cases}
-  \]
-  
+
 \item
-  LÃ¥t \(p\) vara ett udda primtal. Visa att ekvationen
-  \[ x^4 \equiv -1 \mod p\]
-  är lösbar om och endast om \(p \equiv 1 \mod 8\).
+        Sats 9.17 i Rosen säger att kongruensen ovan har en lösning om och endast om
+        \begin{displaymath}
+          (-1)^{\phi(p)/d} \equiv 1 \mod p
+        \end{displaymath}
+        där \(d=\sgd(4,\phi(p)) = \sgd(4,p-1)\).
 
-  \soln
-  Rosen har en sats om \(n\)-ära residyer.
-  
-\item Vad är den största multiplikativa ordningen
-  av ett heltal modulo 99?
-  Ange något sådant heltal av maximal multiplikativ ordning.
+        \(p =8k+1\): Nu blir
+        \(\frac{p-1}{d} = \frac{8k}{4} = 2k\) vilket är jämnt, så
+        \((-1)^{\phi(p)/d} \equiv 1 \mod p\).
 
-  \soln
-  Eftersom \(99=3^2*11\) så ges denna maximala ordning
-  av \(\mgm(\phi(9),\phi(11)) = \mgm(6,10) = 30\),
-  enligt sats i Rosen.
-  Vi ser att redan \(\mathrm{ord}_{99}(2)=30\).
-\item  Visa att \(n^3(n^{18}-1) \equiv 0 \mod 26\)
-  för alla heltal \(n\).
+        \(p =8k+3\): Nu blir
+        \(\frac{p-1}{d} = \frac{8k+2}{2} = 4k+1\) vilket är udda, så
+        \((-1)^{\phi(p)/d} \equiv -1 \mod p\).
+
+        \(p =8k+5\): Nu blir
+        \(\frac{p-1}{d} = \frac{8k+4}{4} = 2k+1\) vilket är udda, så
+        \((-1)^{\phi(p)/d} \equiv -1 \mod p\).
+
+        \(p =8k+7\): Nu blir
+        \(\frac{p-1}{d} = \frac{8k+6}{2} = 4k+3\) vilket är udda, så
+        \((-1)^{\phi(p)/d} \equiv -1 \mod p\).
+
+
+  
 
-  \soln Taget, och förenklat, från en gammal Melkersson-tenta,
-  lösning finns på kurshemsidan.
+\item   Härma lösningen till första uppgiften på tentan som gick
+        2016-06-09; finns på kurshemsidan.
 \end{enumerate}
 
 
diff --git a/TATA54/TENTOR/tenta-TATA54-20240601.log b/TATA54/TENTOR/tenta-TATA54-20240601.log
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diff --git a/TATA54/TENTOR/tenta-TATA54-20240601.tex b/TATA54/TENTOR/tenta-TATA54-20240601.tex
index 775f110..ab76b3e 100644
--- a/TATA54/TENTOR/tenta-TATA54-20240601.tex
+++ b/TATA54/TENTOR/tenta-TATA54-20240601.tex
@@ -1,7 +1,7 @@
 \documentclass[12pt,a4paper]{article}
 %\usepackage{times}
 %\usepackage{euler}
-\usepackage{a4wide}
+%\usepackage{a4wide}
 \usepackage{enumerate}
 \usepackage{amsmath,amssymb,amsthm}
 \usepackage[utf8]{inputenc}
@@ -10,7 +10,8 @@
 \newcommand{\set}[1]{\left\{{#1}\right\}}
 \newcommand{\vek}[1]{\boldsymbol{#1}}
 \newcommand\setsuchas[2]{\left\{\,{#1}\,\vrule\,{#2}\,\right\}}
-\newcommand{\Z}{\boldsymbol{Z}}
+\newcommand{\Z}{\mathbb{Z}}
+\newcommand{\QQ}{\mathbb{Q}}
 \newcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}
 \newcommand{\divides}[2]{{#1} \lvert  {#2}}
 \pagestyle{empty}
@@ -30,10 +31,10 @@ Alla problem ger maximalt 3 poäng. Full poäng kräver fullständig lösning.
 
 \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 \(n\).}}
+{\small{\(\mu\) är Möbiusfunktionen, \(\phi\) är Eulers phi-funktion,
+som räknar antalet multiplikativt inverterbara kongruensklasser modulo \(n\),
+     \(\tau\) räknar antalet positiva delare till heltalet \(n\).
+}}
 
 \medskip
 
@@ -45,10 +46,16 @@ som räknar antalet multiplikativt inverterbara kongruensklasser modulo \(n\).}}
   \end{displaymath}
   och bestäm \textbf{antalet} sådana lösningar modulo 81.
 
+\item Vad är den största multiplikativa ordningen
+  av ett heltal modulo 99?
+  Ange något sådant heltal av maximal multiplikativ ordning.
+
+
 \item   Hitta alla rationella punkter på kurvan
   \begin{displaymath}
     x^2 -7y^2 - 1 = 0
   \end{displaymath}
+  det vill säga alla lösningar \((x,y)\) med \(x,y \in \QQ\).
 
 \item
     Ange explicit två olika punkter på kurvan
@@ -61,27 +68,26 @@ som räknar antalet multiplikativt inverterbara kongruensklasser modulo \(n\).}}
 \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
-    \[
-      \sum_{k=1}^n \tau(k) =
-      \sum_{k=1}^n  \left \lfloor \frac{n}{k} \right \rfloor
-    \]
+        \[
+        h(n) =
+        \sum_{\divides{k}{n}} \tau(k)
+        = \prod_{j=1}^r \binom{a_j + 2}{2}
+        \]
+        då \(n\) har primfaktoriseringen \(n=\prod_{j=1}^r p_j^{a_j}\).
+        Summan löper förstås över \emph{positiva} delare till \(n\).
+    % 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 \(n^3(n^{18}-1) \equiv 0 \mod 26\)
+\item  Visa att \(n^{15} - n^{3} \equiv 0 \mod 36\)
   för alla heltal \(n\).
 \end{enumerate}
 
diff --git a/TATA54/TENTOR/tentaTATA5420240601sketch.org b/TATA54/TENTOR/tentaTATA5420240601sketch.org
index b6a290e..189f78a 100644
--- a/TATA54/TENTOR/tentaTATA5420240601sketch.org
+++ b/TATA54/TENTOR/tentaTATA5420240601sketch.org
@@ -74,6 +74,31 @@ def H_L_tree(f, p: int, r: int):
 
 Parametrisera med lutning, rationell punkt (-1,0)
 
+#+begin_src sage :session
+var('s,t,x,y')
+kurva = x^2 - 7*y^2-1
+linje = y - t*(x+1)
+soln = solve([kurva,linje],[x,y])
+soln
+rp = soln[1]
+rp
+soln2 = solve([linje],[t])
+soln2
+#+end_src
+
+#+RESULTS:
+: (s, t, x, y)
+: [[x == -1, y == 0], [x == -(7*t^2 + 1)/(7*t^2 - 1), y == -2*t/(7*t^2 - 1)]]
+: [x == -(7*t^2 + 1)/(7*t^2 - 1), y == -2*t/(7*t^2 - 1)]
+: [t == y/(x + 1)]
+
+#+begin_src sage :session :results file :file "u2.png"
+epsi = 10^(-1)
+parametric_plot((rp[0].rhs(),rp[1].rhs()), (t,+epsi,1/sqrt(7) -epsi) )
+#+end_src
+
+#+RESULTS:
+[[file:u2.png]]
 
 ** U3
 
@@ -118,6 +143,13 @@ Antingen lösning till Pell x^2-7y^2=1 eller till negativa Pell x^2-7y^2=-1.
 14 20830/7873 20830 7873 -3
 #+end_example
 
+#+begin_src sage :session
+u = 8 + 3*sqrt(7)
+(u^2).expand()
+#+end_src
+
+#+RESULTS:
+: 48*sqrt(7) + 127
 
 ** U4 
 #+begin_src latex
@@ -134,6 +166,23 @@ Antingen lösning till Pell x^2-7y^2=1 eller till negativa Pell x^2-7y^2=-1.
     \]
 #+end_src
 
+#+begin_src sage :session
+number_of_divisors(10)
+def h(n):
+    return sum(number_of_divisors(k) for k in divisors(n))
+[h(n) for n in range(1,20)]
+def g(s):
+    return binomial(s+2,2)
+def gg(n):
+    return product([g(v[1]) for v in factor(n)])
+
+[gg(n) for n in range(1,20)]
+#+end_src
+
+#+RESULTS:
+: 4
+: [1, 3, 3, 6, 3, 9, 3, 10, 6, 9, 3, 18, 3, 9, 9, 15, 3, 18, 3]
+: [1, 3, 3, 6, 3, 9, 3, 10, 6, 9, 3, 18, 3, 9, 9, 15, 3, 18, 3]
 
 ** U5
 #+begin_src latex
@@ -156,6 +205,62 @@ Ange något sådant heltal av maximal multiplikativ ordning.
 
 ** U7
 #+begin_src latex
-  Visa att \(n^3(n^{18}-1) \equiv 0 \mod 26\) för alla heltal \(n\).
+  Visa att \(n^3(n^{18}-1) \equiv 0 \mod 36\) för alla heltal \(n\).
+#+end_src
+
+#+begin_src sage :session
+p,q =2,3
+a,b = 2,2
+m = p^a*q^b; m
+R.<n> = Integers()[]
+f = n^15 -n^3; f
+[(k,f(k)) for k in Integers(m)]
+f.factor()
+euler_phi(m)
 #+end_src
+
+#+RESULTS:
+#+begin_example
+36
+n^15 - n^3
+[(0, 0),
+ (1, 0),
+ (2, 0),
+ (3, 0),
+ (4, 0),
+ (5, 0),
+ (6, 0),
+ (7, 0),
+ (8, 0),
+ (9, 0),
+ (10, 0),
+ (11, 0),
+ (12, 0),
+ (13, 0),
+ (14, 0),
+ (15, 0),
+ (16, 0),
+ (17, 0),
+ (18, 0),
+ (19, 0),
+ (20, 0),
+ (21, 0),
+ (22, 0),
+ (23, 0),
+ (24, 0),
+ (25, 0),
+ (26, 0),
+ (27, 0),
+ (28, 0),
+ (29, 0),
+ (30, 0),
+ (31, 0),
+ (32, 0),
+ (33, 0),
+ (34, 0),
+ (35, 0)]
+(n - 1) * (n + 1) * n^3 * (n^2 - n + 1) * (n^2 + 1) * (n^2 + n + 1) * (n^4 - n^2 + 1)
+12
+#+end_example
+
 * Slut
diff --git a/TATA54/TENTOR/u2.png b/TATA54/TENTOR/u2.png
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HcmV?d00001

-- 
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