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Problem-Solving and Selected Topics in Number Theory Michael Th. Rassias Problem-Solving and Selected Topics in Number Theory In the Spirit of the Mathematical Olympiads ( Foreword by Preda Mihailescu Michael Th. Rassias Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge CB3 0WB, UK [email protected] ISBN978-1-4419-0494-2 e-ISBN978-1-4419-0495-9 DOI10.1007/978-1-4419-0495-9 Springer New York Dordrecht Heidelberg London MathematicsSubjectClassification(2010): 11-XX, 00A07 ©SpringerScience + BusinessMedia,LLC2011 Allrightsreserved. Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. 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Printedonacid-freepaper Springer is part of Springer Science+Business Media (www.springer.com) To my father Themistocles Contents Foreword by Preda Mih˘ailescu................................. ix Acknowledgments ............................................. xv 1 Introduction............................................... 1 1.1 Basic notions ........................................... 1 1.2 Basic methods to compute the greatest common divisor ...... 4 1.2.1 The Euclidean algorithm ........................... 5 1.2.2 Blankinship’s method .............................. 5 1.3 The fundamental theorem of arithmetic .................... 6 1.4 Rational and irrational numbers........................... 8 2 Arithmetic functions....................................... 15 2.1 Basic definitions......................................... 15 2.2 The Mo¨bius function..................................... 16 2.3 The Euler function ...................................... 20 2.4 The τ-function .......................................... 24 2.5 The generalized σ-function................................ 26 3 Perfect numbers, Fermat numbers ......................... 29 3.1 Perfect numbers......................................... 29 3.1.1 Related open problems............................. 31 3.2 Fermat numbers......................................... 32 3.2.1 Some basic properties.............................. 32 4 Congruences............................................... 37 4.1 Basic theorems.......................................... 37 5 Quadratic residues......................................... 51 5.1 Introduction ............................................ 51 viii Contents 5.2 Legendre’s symbol....................................... 56 5.2.1 The law of quadratic reciprocity..................... 62 5.3 Jacobi’s symbol ......................................... 70 5.3.1 An application of the Jacobi symbol to cryptography .. 77 6 The π- and li-functions .................................... 79 6.1 Basic notions and historical remarks ....................... 79 6.2 Open problems concerning prime numbers .................. 82 7 The Riemann zeta function ................................ 83 7.1 Definition and Riemann’s paper ........................... 83 7.2 Some basic properties of the ζ-function..................... 84 7.2.1 Applications ...................................... 95 8 Dirichlet series ............................................ 99 8.1 Basic notions ........................................... 99 9 Special topics..............................................103 9.1 The harmonic series of prime numbers .....................103 9.2 Lagrange’s four-square theorem ...........................112 9.3 Bertrand’s postulate .....................................120 9.4 An inequality for the π-function ...........................129 9.5 Some diophantine equations ..............................137 9.6 Fermat’s two-square theorem..............................143 10 Problems ..................................................147 11 Solutions ..................................................163 12 Appendix..................................................291 12.1 Prime number theorem...................................291 12.2 A brief history of Fermat’s last theorem ....................306 12.3 Catalan’s conjecture .....................................310 References.....................................................317 Index of Symbols ..............................................321 Index..........................................................323 Foreword The International Mathematics Olympiad (IMO), in the last two decades, has become an international institution with an impact in most countries throughout the world, fostering young mathematical talent and promoting a certain approach to complex, yet basic, mathematics. It lays the ground for an open, unspecialized understanding of the field to those dedicated to this ancient art. The tradition of mathematical competitions is sometimes traced back to national contests which were organized in some countries of central Europe already at the beginning of the last century. It is very likely that a slight variationoftheunderstandingofmathematicalcompetitionwouldrevealeven more remote ancestors of the present IMO. It is, however, a fact that the present tradition was born after World War II in a divided Europe when the firstIMO tookplaceinBucharestin1959amongthe countriesofthe Eastern Block. As an urban legend would have it, it came about when a high school mathematics teacherfroma smallRomaniantownbeganto pursue his vision for an organizedevent that would help improve the teaching of mathematics. Sincetheearlybeginnings,mathematicalcompetitionsoftheinternational olympiad type have established their own style of problems, which do not requirewide mathematicalbackgroundandareeasytostate.Theseproblems areneverthelessdifficulttosolveandrequireimaginationplusahighdegreeof original thinking. The Olympiads have reached full maturity and worldwide status in the last two decades. There are presently over 100 participating countries. Accordingly, quite a few collections of Olympiad problems have been published by various major publishing houses. These collections include problems from past olympic competitions or from among problems proposed by various participating countries. Through their variety and required detail of solution, the problems offer valuable training for young students and a captivating source of challenges for the mathematically interested adult. In the so-called Hall of Fame of the IMO, which includes numerous presently famous mathematicians and several Fields medalists, one finds a