主题目录

  • 课程介绍( Course Introduction)

    • University: Massachusetts  Institute  of  Technology

      Instructors:Dr. Dan Ciubotaru

      Course Number:18.104

      Level:Undergraduate

      Course Description

      18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory.

  • 教学大纲(Syllabus)

    • Prerequisite

      The prerequisite for this class is Analysis I (18.100A, 18.100B, or 18.100C). We will emphasize those topics which use methods taught in a typical first semester course in Analysis.

  • 其他教学资源(Other Resources)

    Readings 

    Readings are in the required course textbook:

     Young, Robert M. Excursions In Calculus: An Interplay of The Continuous and The Discrete. Washington, DC: Mathematical Association of America, 1992. ISBN: 0883853175. Some additional readings are linked from this page.


    SES #TopicsREADINGS
    1

    Infinitude of The Primes

    Formulas Producing Primes?

    Infinitude of The Primes

    Text, chapter II (1a), pp. 58-63, possibly complemented by exercise 2; p. 34, exercise 2 (maybe also 1); p. 70.

    Formulas Producing Primes?

    Text, chapter II (1b), pp. 64-69, possibly complemented by exercise 6 (maybe 4,5); p. 70, exercises 13 and 15; p. 71.

    2 Summing Powers of Integers, Bernoulli Polynomials Text, chapter II (2), pp. 74-93; possibly complemented by exercises 2 and 3; p. 95, exercise 11; p. 97, exercises 19 and 20; p. 99.
    3

    Generating Function for Bernoulli Polynomials

    The Sine Product Formula and $\zeta(2n)$

    Generating Function for Bernoulli Polynomials

    Text, pp. 160-161.

    The Sine Product Formula and $\zeta(2n)$

    Text, pp. 345-348.

    4 A Summary of the Properties of Bernoulli Polynomials and More on Computing $\zeta(2n)$
    5 Infinite Products, Basic Properties, Examples (Following Knopp, Theory and Applications of Infinite Series)
    6 Fermat's Little Theorem and Applications Text, pp. 100-110 (without Mersenne Primes) and exercises 13 and 14; p. 117.
    7 Fermat's Great Theorem Text, pp. 110-114, exercise 24; p. 119.
    8 Applications of Fermat's Little Theorem to Cryptography: The RSA Algorithm Reference: Trappe, Washington. Introduction to Cryptography with Coding Theory. Section 6.1, a little of 6.3
    9 Averages of Arithmetic Functions Text, pp. 219-225 with exercises 11, 12 and 13; p. 241.
    10 The Arithmetic-geometric Mean; Gauss' Theorem Text, pp. 231-238; maybe supplemented by some material from Cox, David A. Notices 32, no. 2 (1985) (QA.A5135) and Enseignment Math 30, no. 3-4 (1984).
    11 Wallis's Formula and Applications I Text, pp. 248-254, exercises 9 and 10; p. 263, maybe also exercise 11; p. 264.
    12

    Wallis's Formula and Applications II (The Probability Integral)

    Stirling's Formula

    Wallis's Formula and Applications II (The Probability Integral)

    Exercise 1; pp. 272-273, and the "usual" proof, also consult section 5.2, pp. 267-272 if needed.

    Stirling's Formula

    Exercises 13 and 14; pp. 264-267.

    13 Stirling's Formula (cont.) Exercises 13 and 14; pp. 264-267.
    14 Elementary Proof of The Prime Number Theorem I Following M. Nathanson's "Elementary methods in number theory.": Chebyshev's Functions and Theorems. For a historical account, see D. Goldfeld's Note. (PDF)
    15 Elementary Proof of The Prime Number Theorem II: Mertens' theorem, Selberg's Formula, Erdos' Result

    The original papers can be found on JSTOR:

    Selberg, A. "An Elementary Proof of the Prime-Number Theorem."

    Erdos, P. "On a New Method in Elementary Number Theory Which Leads to an Elementary Proof of the Prime Number Theorem."

    16 Short Analytic Proof of The Prime Number Theorem I (After D. J. Newman and D. Zagier)

    The original papers are on JSTOR:

    Newman, D. J. "Simple Analytic Proof of the Prime Number Theorem."

    Zagier, D. "Newman's Short Proof of the Prime Number Theorem."

    17 Short Analytic Proof of The Prime Number Theorem II: The Connection between PNT and Riemann's Hypothesis

    An Expository Paper:

    Conrey, J. Brian. The Riemann Hypothesis in the "Notices of the AMS". (PDF)

    18 Discussion on the First Draft of the Papers and Some Hints on How to Improve the Exposition and Use of Latex References: Knuth, Larrabee, and S. Kleiman Roberts. (PDF)
    19

    Euler's Proof of Infinitude of Primes

    Density of Prime Numbers

    Text, pp. 287-292, 296-306, and 299-301 (especially Euler's Theorem, pp. 299-301). Also p. 351 in reference [211] (Hardy-Wright) and exercise 4; p. 294.
    20

    Definition and Elementary Properties of Fibonacci Numbers, Application to the Euclidean Algorithm

    Binet's Formula

    Definition and Elementary Properties of Fibonacci Numbers, Application to the Euclidean Algorithm

    Text, pp. 124-130. Exercises 6, 9, and 24, pp. 134-140.

    Binet's Formula

    Morris, pp. 130-132, also the example, "The transmition of information". Exercises 14, 17, and 27, pp. 134-140.

    21

    Golden Ratio

    Spira Mirabilus

    Golden Ratio

    Text, pp. 140-144. Exercises 4 and 9; pp. 154-156, exercise 20; p. 136.

    Spira Mirabilus

    Text, pp. 148-153. Example 1; pp. 159-160 (The Generating Function for Fibonacci Numbers). Exercise 32; p. 138, 21; p. 136.

    22 Final Paper Presentations I
    23 Final Paper Presentations II
    24 Final Paper Presentations III

  • 作业(Assignments)

    Homework is assigned in the required textbook:

     Young, Robert M. Excursions In Calculus: An Interplay of The Continuous and The Discrete. Washington, DC: Mathematical Association of America, 1992. ISBN: 0883853175.