Konstruktion guter ABC-Tripel mit dem LLL-Algorithmus
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Masterarbeit aus dem Jahr 2009 im Fachbereich Mathematik - Zahlentheorie, Note: 2,0, Gottfried Wilhelm Leibniz Universität Hannover (Institul für Algebra, Zahlentheorie und Diskrete Mathematik), Sprache: Deutsch, Abstract: In der Mathematik gibt es eine Reihe zentraler Aussagen, deren Beweis über Jahre brauchte. Zudem gibt es noch heute viele Annahmen, die weder bewiesen noch widerlegt sind. Dazu zählt auch die ABC-Vermutung. Man spricht von einem ABC-Tripel, wenn die Zahlen des Zahlentripels (a; b; c) paarweise teilerfremd sind und zusätzlich die Summe von a und b den Wert von c ergibt mit der Eigenschaft, dass das Radikal aus dem Produkt der drei Zahlen kleiner ist als die grö te der drei Zahlen. Bisher ist unbekannt, ob die Anzahl der Zahlentripel endlich ist. Gilt die ABC-Vermutung, so folgen hieraus eine Reihe weiterer Aussagen, beispielsweise eine schwache Formulierung des letzten Satzes von Fermat, der über 300 Jahre ungelöst war und erst 1993 von Wiles bewiesen wurde. Eine Verschärfung der Aussage über Zahlentripel ergibt sich, wenn zusätzlich die Eigenschaft gut verlangt wird. Von guten Zahlentripeln spricht man, wenn der Quotient aus dem Logarithmus der betragsgrö ten Zahl und dem Logarithmus des Radikals vom Produkt der drei Zahlen grö er als 1,4 ist.
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