UNIVERSITY OF WAIKATO

Department of Mathematics

MATH509-11A (and MATH31911A) – Topics in Number Theory - 2011

Class Web Site
I will be available Tuesday 12th June 2pm-4pm and Thursday 16th June 2pm-4pm for student assistance. The list of topics for test 2(math509) and the number theory part of the examination (math319) is given below. Note that it will not be all bookwork, but some problems like those in the assignments will also be included. Test 2 for math509 will take place in G3.33 on Monday morning 20th June - see the class notice board. The Exam for math319 will be at a corresponding time in a room arranged by the University.

Solutions for assignment 3 are linked below.

There are some small corrections to the notes for pages 172-174 in the new version linked below.

Topics for the test and examination:

All definitions and theorem statements. Proofs of Prop 1,2,3, Thm 1,2, Prop 5,6,7, Thm 3, Prop 9,10,11,12, 14,15,16, Thm 5,6,7,8, Prop 17,18,19,20, Thm 9 + corollary, Thm 11, Prop 24, 25, Thm 13, Prop 26,27, Thm 14, Prop 28, Thm 15, 16, Prop and Thms on p74, p75 and the 2nd on p79, phythag eqn, corollaries to Thm 23, Thm 24.

Box on p98, Thm 26, Return to Pell p117, Props p122, 123, Thm 29, Props p125, 126, Thm 32, Thm 37, Thm 36, Prop p167, Cor p164, Thm 35, Thm 33

Rationale:

For some time now there has been developing within and outside of mathematics a renewed energy and interest in matters relating to number theory. This has come from not only the recent solution to some unsolved long outstanding problems using modern methods, like Wiles’ proof of Fermat’s Last Theorem, but also because of the pressing need for effective encryption in commercial, strategic and personal computer based communications. In addition, the use of the computer has made it possible to explore a much wider domain of number based phenomena than before, leading to new ideas.

Details of the paper content:

The following is a list of the type of topics which might be included, but it is not exhaustive and all topics listed would not necessarily be covered: Theory of prime numbers: fundamental theorem of arithmetic, sieve of Eratosthenes, factoring large numbers into prime factors, computer based probabilistic primality tests, Bertrand’s postulate. Special types of number – Fermat, perfect, etc. Cryptography. The distribution of primes, Chebyshev’s theorem and the prime number theorem, arithmetical functions and mobius inversion, Euler’s summation formula, average orders of arithmetical functions. Diophantine equations: pythagorian triples, Fermat’s last theorem, rational points on elliptic curves, elliptic curve factoring algorithms. Computer implementation. Algebraic number theory: rational, algebraic and transcendental numbers, approximation of irrationals by rationals, continued fractions, Liouville’s transcendental numbers, e and pi are irrational. Unsolved problems in number theory: including the twin primes conjecture, Goldbach’s conjecture, and the Riemann hypothesis. Computer exploration of conjectures. Historical aspects.

Bibliography:

There are a number of texts, most of which should be in the library which should be suitable for additional reading:
  • Elementary Number Theory, David M. Burton (Allyn and Bacon 1976),
  • An Introduction to the Theory of Numbers, Niven and Zuckerman (Wiley 1972)
  • Fundamentals of Number Theory, W J. LeVeque (Addison Wesley 1977)
  • Solved and Unsolved Problems in Number Theory D Shanks (Spartan 1962)
  • An introduction to the Theory of Numbers G H Hardy and E M Wright (Oxford 1938-1999)
  • From Fremat to Minkowski W Scharlau H Opolka, (Springer Verlag 1985)
  • Introduction to Analytic Number Theory, T M Apostol (Springer Verlag 1976)
  • Irrational Numbers, I Niven (Mathematical Assoc of America 1963)
  • Number Theory with Computer Applications, R. Kumanduri and C Romero (Prentice Hall 1998),
  • Number Theory a Programmers Guide, M. Herkammer (McGraw Hill 1999),
  • Primes and Programming, P. Giblin (Cambridge 1993).
  • A primer of analytic number theory, by Jeffrey Stopple, (CUP).

    Web Sources:

  • Paper outline,
  • Number Theory Web,
  • Pages on primes,
  • History of Mathematics,
  • Cryptography,
  • Fermat’s Last Theorem.

    Lecturer

    Timetable

    Lectures (all in G3.33):

    1. Monday 11.00 a.m. - 11.50 a.m.
    2. Tuesday 5.10 p.m. - 6.00 p.m.
    3. Thursday 5.10 p.m. - 6.00 p.m.(normally a tutorial)

    Computer Accounts

    Students in this course are able to use the Senior Computational Mathematics Lab G3.12.

    Texts and Information Sources

    Prescribed Texts:

  • Elementary Number Theory and its applications by Kenneth Rosen, 6th Edition, Addison Wesley.
  • Elementary Number Theory by Peter Hackman, online, 2009, 450 pages.

    On-line lecture notes:

    The text for the on-line lecture notes, Number Theory by Kevin A. Broughan Copyright (C) 1998-2010, may be obtained by clicking on the link below:

    Readings and exercises from the texts for 2011 will go here:

    1. Factorisation: Notes p1-3; Rosen read sections 3.1 3.5. Do Rosen p76 # 1(a)(f), 3,5,11,13,27.
    2. The GCD: Notes p5-10; Rosen read section 3.2, do p99 #1(a),3,5,7,10,15,19,25,31.
    3. Linear equations in Z, perfect numbers: Notes p10-15, Rosen read section 3.7, Do p141 #1(a)(b)(c), 711(a), 21,24.
    4. Perfect numbers, Mersenne primes: Notes p13-19, Rosen read sections 7.2, 7.3, Do page 253 # 1(a)(b), 3, 5(a),9 p266 # 1, 3(a), 7,9,15(a), 17.
    5. Congruences (plus a little on Fermat numbers): Notes p20-27, Rosen read section 4.1, Do p153 # 1(b)(g), 3(a), 5, 7(a), 9(b), 11,13,27.
    6. Moebius function: Read Rosen sections 6.1, 6.3, 7.4, 7.1. Do p222 #1,5,17, p237 #1(c),7,11(a), p245 #3,5,7.
    7. Mobius inversion: Read Rosen section 7.4, Do p274 #1(a), 3,7,13,19,21,23.
    8. Euler summation and the average of the sum of divisors d(n): Notes p39-46,p56.
    9. Chebyshev's approximation to the PNT: Notes p47-54.
    10. Bertrand's postulate: Notes p55-69.
    11. Unsolved problems for prime numbers: Notes p70.
    12. Numbers expressible as sums of 2 squares: Read Rosen section 13.3 p542-545, do page 550 #1(a)(b), 3(a)(b), 5(b),7.
    13. Numbers expressible as the sums of 4 squares. The pythagorian equation. Read Rosen p546-550, Do p551, #9(a), 10,11,17,18. Also read section 13.1 (pythagorian equations), and p619 (Mathematica for number theory). Do p528 #1(a)(b),3,5,11,13.
    14. Diophantine equations, Fermat's last Theorem, Read notes p84-88 and those on Pythagorian equations linked above, Do Rosen p521 sections 13.1,13.2, do page 540 # 1, 3(a)(b),7,9,11,19.
    15. Pell's equation: Read Rosen p557 only Thm 13.12 and Ex 13.11, do p558 # 1(a)(b), 3(a)(b), 4(a)(b) (consult the table in your notes), 5, 9, 11.
    16. Quadratic reciprocity: Notes p121-, Rosen ch11 p415-. Do p425# 1(c),3,5,7,9,11,13(b).
    17. Quadratic reciprocity II: Notes p121- , Rosen section 11.2 p430-, Do p439 # 1(a)(b),5,7(b),10,11.
    18. Continued fractions I: Notes p96-, Rosen Sections 12.2- p481, Do p489 #1(d), 3(a), 5(a), 9.
    19. Continued fractions II quadratic irrationals: Notes p108- Rosen Section 12.4, do p515 #1(a)(b), 3(a), 5(a)(c),9(a)+Sqrt[99], 15(a)(b).
    20. Continued fractions and Pells equation: Notes p115-, Rosen Section 13.4, use C.F.'s to do p559 4(a)(b),5,7.
    21. Numbers rational and irrational: e and pi: Notes p157 (note: we don't really need the two lemmas, something simpler will be given), book by Ivan Niven.
    22. Measure zero, algebraic and transcendental nos: Notes p164-, book Hardy and Wright.
    23. Algebraic and transcendental nos: Notes p171-

    The text of the assignments for 2011:

    1. Assignment 1 due Friday 18th March, solutions,
    2. Assignment 2 due Friday 1st April, solutions,
    3. Assignment 3 due Friday 20th May, solutions,
    4. Assignment 4 due Friday 3rd June.

    Assessment

    General Rules for math509:

    The assessment will consist of two formal tests, worth 20% and 28% of the final mark respectively, and 4 marked assignments each worth 13%. Assignments will be handed out at the tutorial on Thursday and should be handed in through the slot marked 509 under the Mathematics Office reception counter (G3.19) by the following Thursday. Late assignments will be penalized.

    Dates and values of the Assignments and the two tests:

    1. Assignment 1: Out Thursday 10th March, back Friday 18th March, value 13%,
    2. Assignment 2: Out Thursday 24th March, back Friday 1st April, value 13%,
    3. Recess
    4. Test 1: Thursday 5th May in G3.33, value 20%.
    5. Assignment 3: Out Thursday 12th May, back Friday 20th May, value 13%,
    6. Assignment 4: Out Thursday 26th May, back Friday 3rd June, value 13%,
    7. Test 2: in June after the study break, at a date to be announced, value 28%.

    1. A test from 2008,
    2. A test from 2003,
    3. An examination from 2003.

    Professor Kevin Broughan

    3rd June 2011