Notes for Number theory (Fall semester) Sophie Marques August 2014 2 Contents I Arithmetic of Z, divisibility theory 11 1 Interlude on natural numbers, induction and well ordering 13 2 Divisibility 15 2.1 Definition of divisibility . 15 2.2 Some divisibility tests . 15 2.2.1 Divisibility by 10n, 5, 25 ...................... 16 2.2.2 Divisibility by a power of 2 .................... 16 2.2.3 Divisibility by 3 and 9 ....................... 16 2.2.4 Divisibility by 11 .......................... 16 2.2.5 More divisibility test . 17 2.3 G.C.D.andL.C.M.............................. 17 2.4 Prime and coprime numbers . 18 3 Euclidean division, Bezout theorem, linear diophantine equations 21 3.1 Euclidean division . 21 3.2 Euclideanalgorithm ............................ 22 3.3 Bezout'sidentity .............................. 23 3.4 Application: linear diophantine equations. 26 4 The fundamental theorem of arithmetic 29 4.1 Thetheorem................................. 29 4.2 Applications................................. 31 4.3 Primality-testing and factorization . 33 II Arithmetic Functions 37 5 Arithmetic Functions 39 5.1 Definitions, examples . 39 5.2 Euler'sfunction............................... 41 5.3 Convolution, M¨obius inversion . 42 3 4 CONTENTS III Modular arithmetic on Z 45 6 Congruences 47 6.1 Motivation.................................. 47 6.2 Definition and first properties . 48 7 Congruence equations 53 7.1 Congruences and polynomials . 53 7.2 Linear congruences . 54 7.2.1 Simple linear congruences . 54 7.2.2 Simultaneous linear congruences, chinese remainder theorem . 57 7.2.3 Congruences with prime modulus . 61 7.2.4 Congruences with prime power modulus . 64 8 The ring pZ{nZ; `; ¨q, its group of unit Un, applications 67 8.1 Algebraicinterlude ............................. 67 8.2 The ring pZ{nZ; `;:q and its group of Units . 69 8.3 Another proof of the multiplicativity of the Euler function . 72 8.4 Application to cryptography . 74 8.5 Modular arithmetic revisited by algebra . 77 9 Quadratic reciprocity 83 9.1 The legendre symbol . 83 9.2 Euler's Criterion . 83 9.3 The Quadratic Reciprocity Law . 85 9.4 A Lemma of Gauss . 86 9.4.1 A group theoretic proof . 88 9.4.2 Applications . 89 9.4.3 Jacobi symbols . 91 10 Continued fractions 95 10.1Introduction................................. 95 10.2 Continued fractions for quadratic irrationals. 98 10.3Pell'sequation................................ 101 11 Gaussian integers 105 11.1 Basic properties . 105 11.2 Fermat's two square theorem . 107 11.3 Pythagorean triples . 109 11.4 Primes of the form 4n ` 1 ......................... 111 12 Other diophantine equation 113 12.1 Fermat's equation . 113 12.2 Mordell's equation . 115 12.3 The 'abc'-conjecture . 117 12.4 Mordell's conjecture . 118 Introduction The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". The word "arithmetic" (from the Greek, arithmos which means "number") is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic. Arithmetic is the old- est and most elementary branch of mathematics, used very popularly, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine num- bers. The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic. Elementary arithmetic starts with the natural numbers and the written symbols (dig- its) which represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of num- bers, including the contents of a multiplication table to assist with multiplication and division. Elementary arithmetic also includes fractions and negative numbers, which can be represented on a number line. Number theory is devoted primarily to the study of the integers. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophan- tine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic num- ber theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (Diophantine approximation). 5 6 CONTENTS Fixing notations Symbol Meaning @ for all, for every; D there exists (at least one); D! there exists exactly one; s:t: such that; ñ implies; ô if and only if; x P A the point x belongs to a set A x R A the point x does not belongs to the set A N the set of natural number (counting numbers) 1; 2; 3;::: Z the set of all the integers (positive, negative or zero) Q the set of rational numbers R the set of real numbers C the set of complex numbers tx P A : Ppxqu the subset of the elements x in a set A such that the statement Ppxq is true H the empty set, the set with nothing in it x P A means that the point x belongs to a set A or that x is a element of A A Ď BA is a subset of B i.e. any element of A also belongs to B (in symbolic notation: x P A ñ x P B that we use when doing proofs). A “ B the sets A and B contain exactly the same points This statement is equivalent to saying: A Ď B AND B Ď A tpu singleton set (Logically speaking, the "point p" is not the same thing as "the sets tpu whose only element is p.") A X B indicates the intersection of two sets; An element x is in A X B ô x P A AND x P B. Notice A X B “ B X A. A Y B the union of two sets. An element x lies in A Y B ô Either x P A OR x P B (or BOTH). Notice A Y B “ B Y A. 7 8 CONTENTS Symbol Meaning n i“1 Ai “ A1 X ¨ ¨ ¨ X An the set tx : x P Ai f or every iu where Ai are sets n Ai “ A1 Y ¨ ¨ ¨ Y An the set tx : D some i such that x P Aiu where Ai are sets Şi“1 Aα the set tx : x P Aα f or every α P Iu Ťα where Aα are sets and I a set of indexes Ş α Aα the set tx : D some i such that x P Aiu where Aα are sets and I a set of indexes AŤzB the difference set tx : x P A and x R Bu (Note that AzB “ A X Bc. Ac the complement of a set A. Here A is a subset of some larger space X and its complement is the set Ac “ tx P X : x R Au “ XzA. A ˆ B the cartesian product tpa; bq|a P A and b P Bu n A1 ˆ ¨ ¨ ¨ ˆ An “ i“1 Ai the product of the sets Ai tpa1;:::; anq|ai P Aiu 8 Ai the product of the sets Ai tpa1;:::; an;::: q|ai P Aiu i“1 ± Aα the set consisting of all indexed words paαqαPI, where aα P Aα. ±αPI These are the maps φ : I ÑYαPIAα such that φpαq P Aα ± for every index α P I. Mappings φ : X Ñ Y A map φ from a set X to another set Y is an operation that associates x ÞÑ φpxq each element in X to a single element in Y. Unless stated otherwise, mapping φpxq are assumed to be defined for every x P X. If not, De f pφq domain of definition of φ, i.e. the points such that φ is defined. Rangepφq “ φpXq the range of the map φ i.e. tb P Y : Da P X such that b “ φpaqu φpSq the forward image for any subset of S Ď X i.e. tb P Y : Da P S such that b “ φpaqu “ tφpaq : a P Au φ : X ãÑ Y φ is injective or "one-to-one" i.e. if a1 ‰ a2 ñ φpa1q ‰ φpa2q φ : X Y φ is surjective i.e. if φpXq “ Y φ : X » Y φ is bijective i.e. φ is both one-to-one and onto i.e. @b P Y; D!a P X such that φpaq “ b “ φ φ, : X Ñ Y are equal, i.e. they have same action: φpaq “ paq for all a P X. (Ex: px2 ` x4q{p1 ` x2q “ x2.) Γpφq the graph of Γpφq which is a subset of the Cartesian product set X ˆ Y: tpx; yq P X ˆ Y : y “ φpxqu “ tpx; φpxqq : x P Xu pxq ˆ Y "vertical fiber" for each x P X, tpx; yq : y P Yu “ tall points pa; bq P X ˆ Y such that a “ xu Γpφq X ppxq ˆ Yq “ tpx; bqu with b “ φpxq, knowing φ is equivalent with knowing Γpφq. idX : X Ñ X the identity map on X sending each point x P X to x itself. φ´1 : Y Ñ X the inverse map of φ : X Ñ Y such that φ´1pφpaqq “ a, for all a P X and φpφ´1pbqq “ b, for all b P Y φ´1pbq “ the unique element a P X such that φpaq “ b.