TY - RPRT A1 - Koch, Christian A1 - Sultanow, Eldar A1 - Cox, Sean T1 - Divisions by Two in Collatz Sequences N2 - The Collatz conjecture is an unsolved number theory problem. We approach the question by examining the divisions by two that are performed within Collatz sequences. Aside from classical mathematical methods, we use techniques of data science. Based on the analysis of 10,000 sequences we show that the number of divisions by two lies within clear boundaries. Building on the results, we formulate and prove several theorems on the occurrence of cycles and the termination of Collatz sequences. The findings are useful for further investigations and could form the basis for a comprehensive proof of the conjecture. KW - Collatz Conjecture KW - Number Theory KW - Data Science KW - Collatz-Problem KW - Zahlentheorie KW - Data Science Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:92-opus4-6172 ER - TY - RPRT A1 - Koch, Christian A1 - Sultanow, Eldar A1 - Cox, Sean T1 - Divisions by Two in Collatz Sequences: A Data Science Approach (2nd ed.) BT - Working Paper Technische Hochschule Nürnberg 05 May 2020 N2 - The Collatz conjecture is an unsolved number theory problem. We approach the question by examining the divisions by two that are performed within Collatz sequences. Aside from classical mathematical methods, we use techniques of data science. Based on the analysis of 10,000 sequences we show that the number of divisions by two lies within clear boundaries. Building on the results, we develop and prove an equation to calculate the maximum possible number of divisions by two for any given a Collatz sequence. Whenever this maximum is reached, a sequence leads to the result one, as conjectured by Lothar Collatz. Furthermore, we show how many divisions by two are required for a cycle of a specific length. The findings are valuable for further investigations and could form the basis for a comprehensive proof of the conjecture. KW - Collatz Conjecture KW - Divisions by Two KW - Data Science KW - Collatz-Problem KW - Zahlentheorie KW - Data Science Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:92-opus4-6208 ET - 2nd ed. ER - TY - RPRT A1 - Koch, Christian T1 - Falsifizierung des Laplaceschen Dämons mit Hilfe eines Computerspiels BT - Arbeitspapier N2 - Diese Arbeit falsifiziert den Laplaceschen Dämon mit Hilfe eines Computerspiels. Das Spiel ist in der Programmiersprache C++ implementiert und basiert auf deterministischen Regeln. Es führt auf dem verarbeitenden Rechner zu einem probabilistischen Ergebnis. Geht man davon aus, dass der Laplacesche Dämon seinen schnellsten Rechner zur Ausführung des Spiels nutzt, oder es selbst ausführt, kann er die Ergebnisse nicht zu jedem Zeitpunkt vorhersagen. Da der Laplacesche Dämon dadurch definiert ist, dass er die Zukunft vollständig vorhersehen kann, ist er durch das Spiel widerlegt. KW - Laplacescher Dämon KW - Falsifizierung KW - Computerspiel KW - C++ Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:92-opus4-3196 ER -