UNIVERSITY OF WAIKATO

Department of Mathematics

MATH310-08A – Alegebra and Number Theory - 2008

Web page for the Number Theory part

Extra office hours before the final examination: Wednesday and Thursday, 18th and 19th June, 2-5pm in G3.22.

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 Erastosthenes, factoring large numbers into prime factors. 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, algebraic and transcendental numbers, approximation of irrationals by rationals. Proof that e and pi are irrational. Unsolved problems in number theory: including the twin primes conjecture and Goldbach’s conjecture. 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).

    Web Sources:

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

    Lecturer

    Lecture Timetable for the Number Theory section:

    1. Monday 2.10 p.m. - 3.00 p.m. G3.33,
    2. Wednesday 9.00 a.m. - 9.50 a.m. G3.33.

    Computer Accounts:

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

    Texts and Information Sources

    Prescribed Texts:

  • Number Theory, by G. E. Andrews (Dover).

    On-line lecture notes:

    The text for the on-line lecture notes, Number Theory by Kevin A. Broughan Copyright (C) 1998-2003, may be obtained by clicking on the links below:

    Reading and exercises from the text for 2008 will go here:

    1. Factorisation: Andrews Sections 2.1, 2.2.
    2. Primes: Notes p3-7; Andrews Sections 2.3, 2.4.
    3. Linear equations in Z: Notes p7-10; Andrews Sections 6.2,6.3.
    4. Congruences: Notes p10-13; Andrews Interlude p254-259.
    5. Euler Phi function: Notes p13-17; Andrews 6.3, 6.4
    6. Mobius inversion: Notes p17-20; Andrews 6.4.
    7. Bertrand's postulate n <= p < 2n, gaps between primes: Notes p32-42.
    8. Numbers expressible as sums of 2 or 4 squares: Notes p43-48.
    9. Unsolved problems (p38-40) Diophantine equations: Notes p49-51.
    10. DVD BBC Horizon on Fermat's Last Theorem (for graduation day). See also the book (easy to read) called also "Fermat's Last Theorem" by Simon Singh, Fourth Estate 1997 and the notes (including a record of email's) linked elsewhere on this page.
    11. Quadratic reciprocity: Notes p70-77, Andrews ch 9.
    12. Quadratic reciprocity II: Notes p70-77, Andrews Sect 9.3,9.4.
    13. Pell's equation: Notes p52-
    14. Continued fractions I: Notes p56-61, LeVeque ch 9.
    15. Continued fractions III - Pell's equation fund soln: Notes p64-69.
    16. Numbers rational and irrational: e and pi: Notes p73, books by Ivan Niven.
    17. Measure zero, algebraic and transcendental nos: Notes p76-, book Hardy and Wright.
    18. Algebraic and transcendental nos: Notes p80-84.
    19. Applications of the ABC conjecture: p106-109.(It appears Leveque's proof of the restricted Catalan problem is incorrect.)
    20. Formulas for primes: p110-114.
    21. Revision: I will go over the applicable questions in the final examination from 2003.

    The text of the Number Theory assignments for 2008 will go here:

    1. Assignment 1 due Monday 10th March,
    2. Assignment 2 due Friday 28th March,
    3. Assignment 3 due Friday 11th April.
    4. Assignment 4 due Monday 19th May.

    Test 1 2008 Number Theory part

    questions, solutions.

    General Assessment Rules for 314:

    The internal assessment will consist of one formal test, worth 18% of the final mark, and 8 marked assignments (4 for Algebra and 4 for Number Theory) worth a total of 32%. Assignments will be handed out at the lecture on Monday and should be handed in through the slot marked 310 under the Mathematics Office reception counter (G3.19) by the following Monday. Late assignments will be penalized.

    There will be a three hour Final Examination for this paper. The (Internal Assessment):(Final Examination) ratio will be 1:1 or 0:1.

    A copy of the 2003 final examination paper is here.

    Dates and values of Test, Number Theory Assignments and the Final Examination for math310 students:

    1. Assignment 1: Out Monday 3rd March, back Monday 10th March, value 4%,
    2. Assignment 2: Out Monday 17th March, back Wednesday 26th March, value 4%,
    3. Easter
    4. Assignment 3: Out Monday 31st March, back Monday 7th April, value 4%,
    5. Recess
    6. Test: Monday 5th May in G3.33 1 p.m. - 3 p.m., value 18%.
    7. Assignment 4: Out Monday 12th May, back Monday 19th May, value 4%,
    8. Final Examination: in June after the study break, at a date to be announced, value 50%.

    Associate Professor Kevin Broughan

    11th June 2008