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Number Theory and Combinatorics Number Theory and Combinatorics B SURY Stat-Math Unit, Indian Statistical Institute, Bengaluru, India. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the publisher. c Indian Academy of Sciences 2017 Reproduced from Resonance{journal of science education Reformatted by TNQ Books and Journals Pvt Ltd, www.tnq.co.in Published by Indian Academy of Sciences Foreword The Masterclass series of eBooks brings together pedagogical articles on single broad topics taken from Resonance, the Journal of Science Educa- tion, that has been published monthly by the Indian Academy of Sciences since January 1996. Primarily directed at students and teachers at the un- dergraduate level, the journal has brought out a wide spectrum of articles in a range of scientific disciplines. Articles in the journal are written in a style that makes them accessible to readers from diverse backgrounds, and in addition, they provide a useful source of instruction that is not always available in textbooks. The third book in the series, `Number Theory and Combinatorics', is by Prof. B Sury. A celebrated mathematician, Prof. Sury's career has largely been at the Tata Institute of Fundamental Research, Mumbai', and the Indian Statistical Institute, Bengaluru, where he is presently professor. He has contributed pedagogical articles regularly to Resonance, and his arti- cles on Number Theory and Combinatorics comprise the present book. He has also served for many years on the editorial board of Resonance. Prof. Sury has contributed significantly to research in the areas of linear al- gebraic groups over global and local fields, Diophantine equations, division algebras, central extensions of p-adic groups, applications of density theo- rems in number theory, K-theory of Chevalley groups, combinatorial num- ber theory, and generation of matrix groups over rings. The book, which will be available in digital format, and will be housed as always on the Academy website, will be valuable to both students and experts as a useful handbook on Number Theory and Combinatorics. Amitabh Joshi Editor of Publications Indian Academy of Sciences August 2017 iii About the Author B Sury is a Professor of Mathematics at the Indian Statistical Institute in Bangalore since 1999. He was earlier at the Tata Institute of Fundamental Research, Mumbai, where he also got his PhD. Sury's professional interests have been very diverse: from the theory of algebraic groups and arithmetic groups, to algebraic K-theory, and number theory. He has contributed to these areas both through research papers and also through books. Sury enjoys thinking about mathematical problems at all levels, and has taken keen interest in promoting problem solving skills. As a result of his abilities and willingness to shoulder responsibilities both in training stu- dents, and in writing exposition for students, he is universally sought after for any student activity in the country: he has been associated with sev- eral student magazines in the country, perhaps the important of them all, `Resonance' for the last two decades, as well the Ramanujan Math Soci- eties Newsletter. He has been an important member of the Mathematical Olympiad Program of the country. Sury is known to friends and colleagues for his wit and humor, which seems to come almost instantaneously; one can enjoy some of his limericks on his webpage at ISI. The present volume brings together some of the writings of B Sury on Number Theory and Combinatorics which have appeared in `Resonance' during the last two decades. Each of the articles is a masterpiece! I am sure it will be a feast for the intellect of any reader with some mathematical inclination. Dipendra Prasad TIFR, Mumbai v Contents 1 Cyclotomy and Cyclotomic Polynomials: The Story of how Gauss 1 Narrowly Missed Becoming a Philologist 2 Polynomials with Integer Values 25 3 How Far Apart are Primes? Bertrand's Postulate 37 4 Sums of Powers, Bernoulli and the Riemann Zeta function 47 5 Frobenius and His Density Theorem for Primes 55 6 When is a Decimal Expansion Irrational? 63 7 Revisiting Kummer's and Legendre's Formulae 65 8 Bessels Contain Continued Fractions of Progressions 69 9 The Prime Ordeal 75 10 Extending Given Digits to Make Primes or Perfect Powers 87 11 An Irrational Walk and Why 1 is Not Congruent 93 12 Covering the Integers 99 13 S Chowla and S S Pillai: The Story of Two Peerless Indian 103 Mathematicians 14 Multi-variable Chinese Remainder Theorem 127 15 Which Positive Integers are Interesting? 135 16 Counting, Recounting and Matching 149 17 Odd if it isn't an Even Fit! Lighting up Tiling 161 18 Polya's One Theorem with 100 pages of Applications 169 vii Preface Over the last two decades, my expositions in Resonance are on roughly two topics { (i) number theory and combinatorics, and (ii) group theory. In this volume, some of the expositions related to the former topic have been put together. The chapter on the work of Chowla and Pillai is part of an article written in collaboration with R Thangadurai that appeared in Reso- nance. I would like to thank Thangadurai for allowing me to include it here. I have attempted to retain the original write-ups of the articles as much as possible other than carrying out some corrections and (minor) additions. As this is a compendium of articles that appeared in Resonance, at times some material is repeated in different chapters. Further, this makes the compilation somewhat uneven in terms of levels of mathematical maturity required of the reader despite my efforts to ensure uniformity. Concerning my philosophy of mathematics dissemination, I have staunchly held the opinion that many topics in mathematics which are supposedly advanced in nature, can be exposed in a manner understand- able to a motivated undergraduate or postgraduate student. It may be somewhat of a challenge to make it comprehensible while still retaining the essential depth and technicality of the content, but it should nevertheless be possible. Over the years, I have tried to put this in practice. One of the best compliments I have received from Professor K R Parthasarathy, who told a visitor that he had learnt some number theory through these articles. Despite this obvious exaggeration (or rather because of this perhaps!), I was encouraged to keep trying to share some of the beautiful number theoretic and combinatorial ideas I myself enjoyed learning. To my surprise, I have occasionally found references to some of these articles in university courses in various places, and that has been fulfilling. The years 1981-1999 at the Tata Institute of Fundamental Research { where I first saw what excellence means { have had a positive effect on me in terms of being able to appreci- ate good mathematics (irrespective of whether I could originally contribute to it or not). During those years, it was a lot of fun to learn over conversa- tions near the sea-side, at the coffee-table, and during ping-pong sessions from Madhav Nori, Venkataramana, Dipendra Prasad, C S Rajan, Ravi Rao, Raja Sridharan, Amit Roy, R R Simha, Bhatwadekar, Subramaniam, Parameswaran, Nitin Nitsure, Kapil Paranjape, Stephen Lobo, Kumare- san, and many others. In most instances, I would excitedly come up to them with something I had \seen", and they would be willing listeners and teachers. In later years, I have always benefited by talking to students and realizing time and again what they found simultaneously enjoyable as well ix as not easy to understand. In almost all cases, after a meeting or discussion with some students, I felt compelled to write on a certain topic. Inciden- tally, some of the summer projects that students worked on have appeared in Resonance as well and, many of them were rewritten by me { showing my desire to communicate in a particular way. I personally know several colleagues who are much better at writing such expositions but, not many of them seem to find the time to write at this level. I wish they would. Finally, I am indebted to Professor Ram Ramaswamy for suggesting the publication of this volume, for convincing me that it could be useful, and for pushing me to finish this process. B Sury July 2017 x Cyclotomy and Cyclotomic Polynomials: The Story of how Gauss Narrowly Missed Becoming a Philologist A person who does arouse much admiration and a million wows! Such a mathematicians' prince has not been born since. I talk of Carl Friedrich Gauss! Cyclotomy { literally `circle-cutting' { was a puzzle begun more than 2000 years ago by the Greek geometers. In this pastime, they used two implements { a ruler to draw straight lines and a pair of compasses to draw circles. The problem of cyclotomy was to divide the circumference of a circle into n equal parts using only these two implements. As these n points on the circle are also the corners of a regular n-gon, the problem of cyclotomy is equivalent to the problem of constructing the regular n-gon using only a ruler and a pair of compasses. Euclid's school constructed the equilateral triangle, the square, the regular pentagon and the regular hexagon. For more than 2000 years mathematicians had been unanimous in their view that for no prime p bigger than 5 can the p-gon be constructed by ruler and compasses. The teenager Carl Friedrich Gauss proved a month before he was 19 that the regular 17-gon is constructible. He did not stop there but went ahead to completely characterise all those n for which the regular n-gon is constructible! This achievement of Gauss is one of the most surprising discoveries in mathematics.
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