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By Prof. Dr. Otto Forster (auth.)

Dr. Otto Forster ist Professor am Mathematischen Institut der Ludwig-Maximilians-Universität München und Autor der bekannten Lehrbücher research 1-3.

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Factors(10**17 + 1) . 11 103 4013 2 19938_33369 40 §5 Hier ist ohne weitere Untersuchungen nicht klar, ob der letzte Faktor eine Primzahl ist. (In der Tat trifft das zu, vgl. ) Fiir manche Zwecke ist es niitzlich, die Faktoren zur weiteren Bearbeitung in einem Vektor zusammenzufassen. Dies wird durch folgende Funktion geleistet: function factorlist(x: integer): array; var st: stack; q: integer; begin q := 2; yhile q := factor16(x,q) do stack_push(st,q); x := x div q; end; stack_push(st,x); return stack2array(st); end.

Qj = 2 zuriickgibt. B. 100, 1000, 10000) von Zufallszahlen x, y < 10 m , m = 5,10,20,30,40,50,100, den Mittelwert der Anzahl der benotigten Divisionen sowie den Prozentsatz der Divisionen, bei denen qj = 1 bzw. qj = 2 ist, berechnet. Hinweis: Eine (Pseudo- )Zufallszahl < 10 m erhalt man in ARIBAS mit dem Befehl random ( 10**m). Bemerkung. 5% ist, fUr qj = 2 ist der asymptotische Anteil log2(9/8) ~ 17%, siehe [Knu], Chap. 3. 2. 1 mache man Experimente, urn die relative Haufigkeit fUr das Ereignis festzustellen, dass zwei zufallig gewahlte ganze Zahlen x, y < 10 m teilerfremd sind.

Da die Binomial-Koeffizienten fUr 1 ~ k ~ p - 1 durch p teilbar sind und = p, erfUllt das Polynom F das Eisenstein'sche Irreduzibilitats-Kriterium. Daher ist Fund damit auch

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