UNIVERSITY OF WAIKATO

Department of Mathematics

MATH3/4/514-05A – Number Theory - 2005

Paper Outline

Notes:

The final examination for leval 3 students will be on Thursday 23rd June at 9.15am. The 4th and 5th year students should go to G3.33 at 8.50am on that day for the test which will correspond.

I will be available in my office G3.22 Wednesday 15th June and Thursday 16th June 3-5pm to help students.

For Assignment 7: a hint for number 4 - see the mathematica program here. Note that in your answer you should explain how this program works.

For Assignment 7: sections 4/514B should do 1,2,3,4,6. The other questions are optional (i.e. no extra marks for attempting these.)

For Assignment 6 #1 use the curve y^2=x^3-Ax-B. For #2 the points for n=1,2,3 are (1,2), (-1,-4) and (9,-26) respectively. What I should have said however is if P is the first point, verify that 2P is the next and 3P the final point, using the Mathematica programs you wrote for Q1.

A copy of the 2003 final examination paper is here.

Mathematica problems:

Make certain the numlock key is off - else backspace and delete don't work. When printing type in the lpr box -Plw_maths to set the printer. And check the download fonts to printer box - this solves the problem wherein brackets and other special symbols print incorrectly.

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. Applications to Physics: irrationality and chaos, plane crystallography, primes and quantum chaos, small divisor problems for quasiperiodic functions.

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:

  • 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 12 noon - 12.50 p.m.,
    3. Wednesday (Tutorial) 11.00 a.m. - 11.50 a.m. (starts Wednesday 16th March in G3.33)
    4. Friday 9.00 a.m. - 9.50 a.m..

    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).
  • A primer of analytic number theory, by Jeffrey Stopple, (CUP).

    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 from the texts for 2005 will go here:

    1. Factorisation: Notes p1-3; Stopple Preface, Section 2.1; Andrews Sections 2.1, 2.2.
    2. Primes: Notes p3-7; Stopple Section 2.2; Andrews Sections 2.3, 2.4.
    3. Linear equations in Z: Notes p7-10; Stopple Section 2.1, 2.2; Andrews Sections 6.2,6.3.
    4. Congruences: Notes p10-13; Stopple Section 4.1,4.2,4.3,5.2,6.1; Andrews Interlude p254-259.
    5. Euler Phi function: Notes p13-17; Stopple Section 2.3; Andrews 6.3, 6.4
    6. Mobius inversion:Notes p17-20; Stopple 2.3; Andrews 6.4.
    7. Dirichlet and Euler summation: Notes p21-28; Stopple p64-76; Andrews 15.2.
    8. Chebyshev's approximation to the PNT: Notes p28-32; Stopple sect 5.2.
    9. Bertrand's postulate: Notes p32-42; Andrews p110-112.
    10. Unsolved problems (p38-40) Diophantine equations: Notes p49-51.
    11. Quadratic Fermat, Pell's equation: Notes p52-.
    12. Numbers expressible as sums of 2 or 4 squares: Notes p43-48, Andrews ch 11.
    13. Quadratic reciprocity: Notes p70-77, Andrews ch 9, Stopple sect 11.3.
    14. Quadratic reciprocity II: Notes p70-77, Andrews Sect 9.3,9.4.
    15. Continued fractions I: Notes p56-61, LeVeque ch 9.
    16. Continued fractions II - quadratic irrationals: Notes p60-66.
    17. Continued fractions III - Pell's equation fund soln: Notes p64-69.
    18. Numbers rational and irrational: e and pi: Notes p73, books by Ivan Niven.
    19. Measure zero, algebraic and transcendental nos: Notes p76-, book Hardy and Wright.
    20. Algebraic and transcendental nos: Notes p80-84,RSA.
    21. The group of rational points on an elliptic curve: Notes p78-83. Book: Rational points on elliptic curves by Siverman and Tate, Springer.
    22. Elliptic curves mod p: Notes p83-.85
    23. Congruent number problem and elliptic curve y^2=x(x-n)(x+n) p84-88.
    24. Factoring using Pollard's p-1 and Lenstra's elliptic curve methods p100-105.
    25. Applications of the ABC conjecture: p106-109.
    26. Formulas for primes: p110-114.
    27. Formulas for primes II: p112-114.
    28. Axiom D: p115-118.
    29. Intro to partitions - making change and crazy die: p102-104.Andrews 3.4 and ch 12,13.
    30. Euler's pentagonal number theorem, THE partition function p(n). Andrews 14.1, 14.2.
    31. A recurrence and upper bound for p(n).

    The text of the assignments for 2005 will go here:

    1. Assignment 1 (revised) due Wednesday 16th March, solutions.
    2. Assignment 2 due Wednesday 23rd March, solutions.
    3. Assignment 3 due Wednesday 6th April, solutions.
    4. Assignment 4 due Wednesday 11th May.
    5. Assignment 5 due Wednesday 18th May.
    6. Assignment 6 due Wednesday 25th May.
    7. Assignment 7 due Wednesday 1st June.
    8. Assignment 8 due Friday 10th June.

    Assessment

    General Rules for 314:

    The internal assessment will consist of one formal tests, worth 26% of the final mark, and regular marked assignments worth a total of 24%. Assignments will be handed out at the lecture on Wednesday and should be handed in through the slot marked 314 under the Mathematics Office reception counter (G3.19) by the following Wednesday. 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.

    Dates and values of Tests, Assignments and the Final Examination for math314 students:

    1. Assignment 1: Out Wednesday 9th March, back Wednesday 16th March, value 3%,
    2. Assignment 2: Out Wednesday 16th March, back Wednesday 23rd March, value 3%,
    3. Assignment 3: Out Wednesday 23rd March, back Wednesday 6th April, value 3%,
    4. Test: Friday 15th April in G3.33 9 a.m. - 10 a.m., value 26%.
    5. Recess
    6. Assignment 4: Out Wednesday 4th May, back Wednesday 11th May, value 3%,
    7. Assignment 5: Out Wednesday 11th May, back Wednesday 18th May, value 3%,
    8. Assignment 6: Out Wednesday 18th May, back Wednesday 25th, value 3%,
    9. Assignment 7: Out Wednesday 25th May, back Wednesday 1st June, value 3%,
    10. Assignment 8: Out Wednesday 1st June, back Wednesday 8th June, value 3%,
    11. Final Examination: in June after the study break, at a date to be announced, value 50%.
    For 414 and 514 students: The assignments and test dates are the same as the above. Each assignment will be worth 6% and there will be two tests, each worth 26% of the final mark. The second test will be held at the time of the final examination for 314.

    Associate Professor Kevin Broughan (Convenor)

    16th June 2005