Number theory is concerned with the properties of the natural numbers: 1, 2, 3, .... During the seventeenth and eighteenth centuries, number theory became established through the work of Fermat, Euler and Gauss. With the hand calculators and computers of today the results of extensive numerical work are instantly available and the road leading to their discoveries may be traversed with comparative ease. Now in its second edition this book consists of a sequence of exercises that will lead readers from quite simple number work to the point where they can prove algebraically the classical results of elementary number theory for themselves. A modern high-school course in mathematics is sufficient background for the whole book, which, as a whole, is designed to be used as an undergraduate course in number theory to be pursued by independent study without supporting lectures. A pathway into number theory R. P. BURN A pathway into number theory Second edition CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521575409 © Cambridge University Press 1982, 1997 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1982 Reprinted 1989, 1992, 1994 Second edition 1997 A catalogue record for this publication is available from the British Library ISBN-13 978-0-521-57540-9 paperback ISBN-10 0-521-57540-0 paperback Transferred to digital printing 2006 xi xiii 1 1 7 9 13 13 14 14 16 22 22 27 33 36 37 38 39 48 48 53 53 54 55 56 61 64 66 67 68 70 79 79 81 84 87 88 89 97 97 100 101 102 108 108 110 119 119 123 126 126 127 129 130 140 140 141 145 147 147 148 154 154 158 162 164 165 168 169 170 171