From 4609055084e27178ead6d9f0ecc40dcab620589e Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 29 Sep 2025 21:04:52 +0200 Subject: [PATCH] [TI] Finish up kolmogorov-complexity section --- .../kolmogorov-complexity.tex | 11 +++++++++++ semester3/ti/ti-summary.pdf | Bin 802259 -> 817902 bytes semester3/ti/ti-summary.tex | 7 ++++++- 3 files changed, 17 insertions(+), 1 deletion(-) diff --git a/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex b/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex index a1962fe..0a36369 100644 --- a/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex +++ b/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex @@ -88,3 +88,14 @@ Die Annäherung von $\text{Prim}(n)$ and $\frac{n}{\ln(n)}$ wird durch folgende Für jedes $i \in \N - \{ 0 \}$ sei $q_i$ die grösste Primzahl, die $n_i$ teilt. Dann ist die Menge $Q = \{ q_i \divides i \in \N - \{ 0 \} \}$ \end{lemma} + +Lemma 2.6 zeigt nicht nur, dass es unendlich viele Primzahlen geben muss, sondern sogar, dass die Menge der grössten Primzahlfaktoren einer beliebigen unendlichen Folge natürlicher Zahlen mit nichttrivialer Kolmogorov-Komplexität unendlich ist. + +\begin{theorem}[]{Untere Schranke für Anzahl Primzahlen} + Für unendlich viele $k \in \N$ gilt + \begin{align*} + \text{Prim}(k) \geq \frac{k}{2^17 \log_2(k) \cdot (\log_2(\log_2(k)))^2} + \end{align*} +\end{theorem} + +Der Beweis hierfür ist sehr ausführlich ab Seite 42 im Buch erklärt diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 21bfb2744bfac54d2a43ad116d6428f47f8739cb..e441837d99c093e3efff1eb05332a5ae01b5457c 100644 GIT binary patch delta 69635 zcmZUaQ*19x)URvXwr$(CZQJc``>$=g-L-Aoc6V*tKJWLP)8oeq2riw45Xl8g*O4Jc99)m#!n@}H|)A3MTh+5WWf69v~%>tEuF-duiBm`OhtIelQASS)kyRiB~e&TV7yctj&;j>8U)W6!Q28NI<00jOy2v+;)r 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+++ b/semester3/ti/ti-summary.tex @@ -32,14 +32,19 @@ \begin{center} HS2025, ETHZ\\[0.2cm] \begin{Large} - Summary of the Book + Summary of the book \color{MidnightBlue}\fbox{\href{https://link.springer.com/book/10.1007/978-3-658-06433-4}{Theoretische Informatik}}\color{black} \end{Large}\\[0.2cm] + by Prof. Dr. Juraj Hromkovic \end{center} \newpage \printtoc{Orange} +\begin{scriptsize} + \textit{Note: Definitions, Lemmas, etc are often 1:1 copies from the book or paraphrased (as I did not find an easier way of stating them)} +\end{scriptsize} + \newpage