Topics in Number Theory An Olympiad Orie

 

TOPICS IN NUMBER THEORY: An Olympiad-Oriented Approach 

Mas asu um Bi Bill lla al

Amir Hos osse sein in Par arv vard rdii

Sample Chapters Prepared for Academia.com September 2018

 

Dedicated to

Our regular studies, without which, we could have finished this book long time ago. Fermat, the father of modern number theory. Euler, without whom number theory probably wouldn’t be so rich today. Riemann, who made huge contributions to analytic number theory. Ramanujan, the mathematician of mathematicians. Paul Erdős, the man who loved only numbers.

 

Note for Academia Users •  This file is incomplete. We will update the sample chapters regularly and you can always find the latest version of these two chapters from

The password is   Riemann.

•  Paperback version on Amazon: •   The website of the book always contains the updates, errata, and new free materials teria ls that we regula regularly rly post on social media media.. Excl Exclusiv usivee olymp olympiad iad problem-s problem-sets ets will be released in the website in the near fuure:

•   Facebook page:   . Thiss is our mai Thi main n pag pagee for fe feedb edbac ack k and rev review iewss on the bo book. ok. Fee eell fre freee to post any kind of review, comment, solution, or suggestion you have about this book. Amir Hossein Hossein will crea create te a forum for the bo book ok on AoP AoPS S so ev every eryone one can p post ost their solutions to be added to the book later on. •  If you find any mistake or typo in the book, we would be happy to hear it at Facebook or   .

Copyright Notice The cover shows a part of the famous paper of Bernhard Riemann in 1859 in Number Theory Theor y. It was taken from the web website site of Clay Mathem Mathematics atics Insti Institute tute.. You can read as well as the   of this won wonderf derful ul paper by David Davi d Wilki Wilkins ns in their websi website: te:

This file was prepared by Masum Billal and Amir Hossein Parvardi, as a gift to all the kind people who were looking forward to reading our book and is intended for non-commercial purposes only. Copyright  2018 TopicsInNumberTheory.com Masum Billal and Amir Hossein Par Parv vardi All rights reserved.   c

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First Words I wou would ld like to hav havee a few word wordss b before efore divin divingg into the discu discussion. ssion. First of all, from my personal experience, I have found that there is a common practice to learn1 by learning a lot of theories and then investigating how those theories are used to solve problems. As our primary audience would be students who are looking to get into mathematical olympiads, I highly discourage this. Please do not take number theory for a collection of theories just because the word theory is literally juxtaposed with it. That being said, one could could sa say y tha thatt our book its itself elf is a col collec lectio tion n of a lot of the theore orems ms as we well. ll. Sad Sadly ly,, that is partially true for multiple reasons even though it was not our intention at all. When I first thought about writing this book, my intention was to make students realize that they do not need to know a lot of theorems in order to be able to solve problems. My primary target was to build confidence about this claim. But as we kept writing we had to increase the pace since we had to cover a lot (and that was discarding a lot of contents which we thought would ask for even discussion or we just felt lazy about it), we had to increase the pace. I would like to address one more concern. Originally, my plan was to make this book a series of   5  v  volumes olumes,, this being the first one. In those volume volumess I wan wanted ted to discuss a lot of topics such as special numbers like  egyptian fraction  or even interesting numbers like  abundant number or deficient number  and their properties etc or crucial topics such as  Diophantine equations . You will notic noticee tha thatt we hav havee lef leftt a lot of impo importa rtant nt topi topics cs like those out of this book. The reason is, I quickly realized I can hardly finish writing this first volume, and if I wanted to complete the series we will probably have to keep writing my whole life. So, I had to discard a lot of contents and make the book concise which resulted in squeezing in a lot of contents in a few hundreds of pages. Finally, I would like to thank Amir to join me in this project. At one point I stopped writing writi ng the book. If he had not agreed to be a co-author co-author,, this bo book ok woul would d have pro probably bably not been completed completed at all. More so because he agreed to follow follow the style I want wanted ed to write in even when we had objection from reputed publisher like   Springer . Masum Billal 

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In the past three years, we have have alwa always ys been wor worried ried about this bo book. ok. It’s been a long and tedious job to manage everything and edit all we had written a very long time ago. After I studied higher level number theory concepts, there were times that I found errors/typos in my previous drafts for the book. And that sometimes happened two or more times in a short period, and so, it was getting anno annoying. ying. Any Anyhow how,, we managed to finish it at this time of the summer. It is now 5:36 AM, Tuesday, July 17, 2018 that I’m writing this. You can imagine how crazy this process has made me! BUT!   a wise man once said “high quality math books are written in a series of 

editions in long term,” and we are happy to hear that. Man editions Many y of our frien friends ds helped us during the way to finish this book, as mentioned in the Acknowledgement, and we are so proud to have such friends. I wouldn’t be able to finish my part in this book if I didn’t have the support of  Ghobadipasha. Sh my wonderful, beautiful, and lovely wife  Nadia Ghobadipasha Shee ga gav ve me hope to choose mathematics and always believed in me. She was my only one in the hardest days of life.  Peyman an Nasehpou Nasehpour r, whom I knew from the very first semester of my Professor  Peym undergraduate studies in elelctrical englineering at University of Tehran, helped me a lot in the process of enhancing my mathematical abilities to change my field to mathematics (number theory) for my master’s. He is an inspiration to me, and a great colleague colle ague when it comes to teamw teamwork. ork. I’m looking forw forward ard to wor working king with him more ofter.

The idea behind the lattices in the  cover  is a geometric proof of law of quadratic reciprocity. Our friends found other interpretations of it such as Pick’s theorem (which is not the case here) or the sum of positive integers up to   n, which you will realize is also not always true (for all primes   p, q )).. Sec Sectio tion n (   ) is ded dedica icated ted ttoo this pr proof oof and investig inv estigates ates why it is true using count counting ing the latti lattice ce (grid) points. When we picked picked this idea for the cover, we chose quadratic reciprocity because its proof is geometrically visual vis ual and ind indee eed d ve very ry beau beautif tiful. ul. Our hope wa wass to mak makee the read reader er cur curiou iouss beca because use the design looks familiar. familiar. Sta Stay y excite excited d until yo you u read that proof! Or, may maybe, be, go read it if you you have have the prere prerequi quisit sitee kno knowle wledge dge and the puzz puzzle le is boggl boggling ing your mind mind.. I remember I found the idea of the proofone, in one of was Kenneth Rosen’s booksIon discrete mathematics, but I’m not sure which as it a longH.time ago when wrote it. I A used TikZ package for L TEXto write the codes and generate the graphs. There are so many wonderful things to learn in this book. I hope you enjoy it! Amir Hossein Parvardi, University of British Columbia, Vancouver, BC, Canada, July 2018.

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Acknowledgement Here is a list of all the kind people who helped us review, edit, and improve this book. We could not decide who to thank more, hence an alphabetically (last name) ordered list: 1. Thanks to  Ali Amiri, a kind friend who helped us in the cover design. 2. Thanks to   AnréC  from TeX.StackExchange who wrote the code for figure (   ) in base conversion. 3. We are thankf thankful ul to  Arta Khanalizadeh  for designing the cover of the book. 4. Thanks to  Kave Eskandari  for reading the whole book and commenting on the general points. He caught a good mistake in chapter   . 5. Chee Cheers rs to our mathe mathematic matical al friend  Leonard Mihai C. Giugiuc  from Romania who gave us positive and constructive feedback on the book. He also wrote us a wonderful review in the website. Valentio Iverson  for proof-reading chapters   and   , and poin 6. Thanks to   Valentio pointin tingg out the typo in figure (   ).

7. Tha Thanks nks to Aditya Khurmi  for reading the book and giving us positive feedback. Hesam Korki Korki’s comments on chapter   . He m 8. We appre appreciate ciate   Hesam men enti tion oned ed a ffew ew very important typos, including grammatical. He also mentioned a mathematical change in that chapter, which was very helpful. Peyman Nasehpour  sent us the beautiful problem (   ) an 9. Profe Professor ssor   Peyman and d an amazin ama zingg sol soluti ution on usi using ng Pri Prime me Numbe Numberr The Theore orem. m. He also ga gave ve us pre prett tty y use use-ful comments on chapter   . He als alsoo int introdu roduced ced iin n sect section ion (   ) the amica amicable ble numbers to us with a brief historical note on it. Nakagawa’s comments on chapters   and   . He was on onee 10. We appreciat appreciatee   Kenji Nakagawa of the first people who read and reviewed these two chapters when we put them

on the website of the book. Mohammadamin Nejatbakhshes Nejatbakhshesfahani fahani, Iran National 11. We are thankful to   Mohammadamin Olympiad gold medalist (2010) and winner of gold medals at IMS and IMC, who honored us toread and reviewe the whole book and gave us really instructive comments. 12. We wo would uld lik likee to thank   Nur Muhammad Shafiullah Mahi  for his efforts to make this book better. Greg Martin Martin, a faculty member at the 13. We are honored to thank Professor   Greg Mathematic Mathe maticss Depar Departmen tmentt of Univ Universit ersity y of British Colum Columbia. bia. He happens to be Amir Hossein’s Hossein’s Master Master’s ’s supervisor supervisor.. He kindly revi reviewe ewed d a prin printed ted draft of the book and emailed us over 10 major points to correct in the book. We do appreciate his advice on improving the whole context of the book.

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14. We are thankfu thankfull to  Sohrab Mohtat  for his comments on chapter   . Thanks to him,, we av him avoid oided ed a fat fatal al mista mistake ke in the b begi eginni nning ng of the book. He also wrot wrotee a very useful and detailed review for our book in the website. 15. We are thank thankful ful to Aditya Guha Roy  who reviewed the whole book and caught a few LaTeX typos, generalized lemma (   ), fixing problems in chapter   . Aditya wrote an amazing, educative review in our website. Singhal  carefully reviewed and proof-read the whole first part of the 16.   Navneel Singhal

very ery constructiv constructivee comments. We are thankful book (chapters   to   ) and gave us v to him. 17. Tha Thanks nks ttoo  Amin Soofiani, who is a Master’s student of mathematics at University of British Columbia, we noticed there was a mistake in theorem (   ). He did a perfect, precise, and detailed review on chapter   . 18. We are than thankfu kfull to   Sepehr Yadegarzadeh  for informing us about the correct grammatical ical and vocabular vocabulary yp poin oints. ts. umlaut    for Möbius among other grammat

1

umlaut : a mark (¨) used over over a vowel vowel,, as in German or Hungarian, Hungarian, to indicate indicate a different different vowel vowel quality, usually fronting or rounding.

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Contents 15

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Notations •   N, N0 ,   Z, Q, R, P,   and and   C  are the sets of positive integers, non-negative integers, integers, rational numbers, real numbers, primes and complex numbers, respectively. • |a|  denotes the absolute value of   a  for any real number   a. •   min(a, b)  is the minimum of   a  and   b  a  and nd max(  max(a, b)  is the maximum of   a  and   b. •   (a, b)  and   [a, b]  are greatest common divisor and least common multiple of   a   and b  respectively. •   a|b  means   b  is divisible by   a  without any remainder. •   a  ⊥  b  means  means   (a, b) = 1. •   p  is the   nth prime. n

•   π (x)  is the number of primes less than or equal to the real number   x. •   ϕ(n)  is the number of positive integers less than   n  which are coprime to   n. •   d(n)  is the number of divisors of   n. •   σ (n)  is the sum of divisors of   n. •   φ(n)  is the  Euler function . •   µ(n)  is the  Mőbius function . •   λ(n)  is the  Carmichael function  •   v (a)  is the largest non-negative integer   α  so that   n |a  but   n α

n

+1

α

   a.

•   n! = 1 · 2 . . . n. • •

  is the  n k

  binomial coefficient  indexed by non-negative integers   n  and   k .

a  is the  Legendre symbol  for integer   a  and prime   p.  p

• x  is the largest integer not greater than   x. • x  is the smallest integer integer not less than   x.