An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery PDF

By Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery

ISBN-10: 0471625469

ISBN-13: 9780471625469

The 5th variation of 1 of the normal works on quantity concept, written by means of internationally-recognized mathematicians. Chapters are rather self-contained for larger flexibility. New gains comprise multiplied therapy of the binomial theorem, options of numerical calculation and a bit on public key cryptography. includes a superb set of difficulties.

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Extra info for An Introduction to the Theory of Numbers

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April 1780 ” In diesem Abschnitt werden die nat¨ urlichen Zahlen 1, 2, 3, . . als gegeben angenommen. Sp¨ ater, n¨ amlich im letzten Abschnitt dieses Kapitels, wird dann behandelt, wie sie sich aus bestimmten Grundannahmen (allerdings nicht aus dem Nichts) herleiten lassen. Es wird vorausgesetzt, dass dem Leser das Rechnen mit nat¨ urlichen, ganzen, rationalen und reellen Zahlen und die Kleioßer-Beziehung ≥“ oder >“ nicht nerbeziehung ≤“ oder <“ sowie die Gr¨ ” ” ” ” unbekannt sind. F¨ ur nat¨ urliche Zahlen wird diese Beziehung im Folgenden allerdings gekl¨ art.

Xn−1 + xn ) = x + x2 + . . + xn + xn+1 − (1 + x + . . + xn−1 + xn ) = xn+1 − 1 , 44 2. Nat¨ urliche Zahlen womit gezeigt ist, was gezeigt werden sollte. Wegen der verwendeten Punkte (und der damit verbundenen Schwierigkeiten) ist die Umformung allerdings nicht ganz befriedigend. Nat¨ urlich soll die Verwendung von Punkten in Formeln nicht generell verboten und ein unbedingter Formalismus gefordert werden. Doch darf bei ihrer Verwendung kein Zweifel bestehen, was an der Stelle eigentlich zu stehen hat.

Man pr¨ uft Folgendes: 1. Stimmt die Behauptung f¨ ur n = 1 ? 2. Falls die Behauptung f¨ ur ein n richtig ist, kann man folgern, dass sie auch f¨ ur n + 1 stimmt? Sind beide Pr¨ ufungen erfolgreich verlaufen, so muss die Behauptung f¨ ur alle nat¨ urlichen Zahlen richtig sein. F¨ ur die Begr¨ undung dieser Aussage benutzt man die (intuitiv wenig erstaunliche) Tatsache, dass zwischen einer beliebigen urliche Zahl liegt. nat¨ urlichen Zahl n und n + 1 keine weitere nat¨ Ganz ausf¨ uhrlich kann man sich den Vorgang des Testens so vorstellen: Da n¨ amlich nach (1) die Behauptung f¨ ur n = 1 richtig ist, so muss sie nach (2) auch f¨ ur n = 2 richtig sein.

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An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery


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