Elementary Number TheoryOur intention in writing this book is to give an elementary introduction to number theory which does not demand a great deal of mathematical back ground or maturity from the reader, and which can be read and understood with no extra assistance. Our first three chapters are based almost entirely on Alevel mathematics, while the next five require little else beyond some el ementary group theory. It is only in the last three chapters, where we treat more advanced topics, including recent developments, that we require greater mathematical background; here we use some basic ideas which students would expect to meet in the first year or so of a typical undergraduate course in math ematics. Throughout the book, we have attempted to explain our arguments as fully and as clearly as possible, with plenty of worked examples and with outline solutions for all the exercises. There are several good reasons for choosing number theory as a subject. It has a long and interesting history, ranging from the earliest recorded times to the present day (see Chapter 11, for instance, on Fermat's Last Theorem), and its problems have attracted many of the greatest mathematicians; consequently the study of number theory is an excellent introduction to the development and achievements of mathematics (and, indeed, some of its failures). In particular, the explicit nature of many of its problems, concerning basic properties of inte gers, makes number theory a particularly suitable subject in which to present modern mathematics in elementary terms. 
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Contents
II  1 
III  2 
IV  7 
V  12 
VI  13 
VII  16 
VIII  19 
IX  25 
XLV  148 
XLVI  152 
XLVII  154 
XLVIII  157 
XLIX  162 
L  163 
LI  165 
LII  166 
X  30 
XI  32 
XII  35 
XIII  37 
XIV  46 
XV  52 
XVI  57 
XVII  59 
XVIII  62 
XIX  65 
XX  72 
XXI  78 
XXII  82 
XXIII  83 
XXIV  85 
XXV  92 
XXVI  96 
XXVII  97 
XXVIII  99 
XXIX  103 
XXX  106 
XXXI  108 
XXXII  110 
XXXIII  113 
XXXIV  116 
XXXV  117 
XXXVI  119 
XXXVII  120 
XXXVIII  123 
XXXIX  130 
XL  135 
XLI  138 
XLII  140 
XLIII  143 
XLIV  146 
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Common terms and phrases
absolutely convergent algebraic apply arithmetic functions calculate Carmichael number Chapter Chinese Remainder Theorem complex numbers composite compute converges absolutely Corollary 5.7 cyclic group deduce define denote Dirichlet series distinct primes divisible equation equivalent Euclid's algorithm Euler's Example Exercise exponent factors Fermat numbers Fermat primes finite follows Gaussian integers gcd(a gives greatest common divisor hence identity implies induction infinitely many primes instance irreducible least Lemma linear congruence mathematics Mersenne Mersenne primes method Minkowski's Theorem mod 9 mod p2 moduli mutually coprime nonzero number system number theory odd prime pair perfect square prime numbers prime q primepower factorisation primitive root mod proof of Theorem prove Pythagorean triple quadratic residue rational numbers result satisfying set of residues Similarly simultaneous congruences single congruence class solve square roots squarefree subgroup Theorem 4.3 unique unit mod vol(X