By James J. Tattersall
Meant to function a one-semester introductory path in quantity idea, this moment version has been revised all through. specifically, the sector of cryptography is highlighted. on the middle of the publication are the main quantity theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler. moreover, a wealth of recent routines were incorporated to totally illustrate the houses of numbers and ideas built within the textual content. The booklet will function a stimulating creation for college students new to quantity idea, despite their history. First variation Hb (1999) 0-521-58503-1 First version Pb (1999) 0-521-58531-7
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Paulo Ribenboim behandelt Zahlen in dieser außergewöhnlichen Sammlung von Übersichtsartikeln wie seine persönlichen Freunde. In leichter und allgemein zugänglicher Sprache berichtet er über Primzahlen, Fibonacci-Zahlen (und das Nordpolarmeer! ), die klassischen Arbeiten von Gauss über binäre quadratische Formen, Eulers berühmtes primzahlerzeugendes Polynom, irrationale und transzendente Zahlen.
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Additional info for Elementary Number Theory in Nine Chapters, Second Edition
Gottfried Leibniz and Pietro Mengoli determined the sum of the reciprocals of the triangular numbers, 1 X 1 1 ¼ 1 þ 13 þ 16 þ 10 þ ÁÁÁ : t n¼1 n What does the sum equal? 27. A lone reference to Diophantus in the form of an epitaph appears in the Greek Anthology of Metrodorus, a sixth century grammarian. R. Paton, ‘This tomb holds Diophantus. Ah, how great a marvel! the tomb tells scientiﬁcally the measure of his life. God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this, he clothed his cheeks with down; he lit him the light of wedlock after a seventh part, and ﬁve years after his marriage he granted him a son.
Given two numbers of opposite parity, show that their sum and difference are odd. 20 The intriguing natural numbers 2. Nicomachus generalized oblong numbers to rectangular numbers, which are numbers of the form n(n þ k), denoted by r n, k, where k > 1 and n . 1. Determine the ﬁrst ten rectangular numbers that are not oblong. 3. Prove algebraically that the sum of two consecutive triangular numbers is always a square number. 4. Show that 9tn þ 1 [Fermat], 25tn þ 3 [Euler], and 49tn þ 6 [Euler] are triangular.
5. 2 10 ... ... 2 Sequences of natural numbers 25 natural numbers is a Galileo sequence with ratio 3. If a1 , a2 , a3 , . . is a Galileo sequence with ratio k, then, for r a positive integer, ra1 , ra2 , ra3 , . . is also a Galileo sequence with ratio k. A strictly increasing Galileo sequence a1 , a2 , a3 , . . , with ratio k > 3, can be generated by the recursive formulas !! (k þ 1)an À 1 a2 nÀ1 ¼ 2 and a2n ¼ (k þ 1)an 2 !! þ 1, for n > 2, where a1 ¼ 1, a2 ¼ k, for k > 2, and ½½x denotes the greatest integer not greater than x.