| 1 |
Infinitude of The Primes
Formulas Producing Primes?
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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.
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| 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 $\zeta(2n)$
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Generating Function for Bernoulli Polynomials
Text, pp. 160-161.
The Sine Product Formula and $\zeta(2n)$
Text, pp. 345-348.
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| 4 |
A Summary of the Properties of Bernoulli Polynomials and More on Computing $\zeta(2n)$ |
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| 5 |
Infinite Products, Basic Properties, Examples (Following Knopp, Theory and Applications of Infinite Series) |
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| 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
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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.
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| 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."
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| 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."
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| 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)#
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| 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
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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
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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.
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| 21 |
Golden Ratio
Spira Mirabilus
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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.
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| 22 |
Final Paper Presentations I |
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| 23 |
Final Paper Presentations II |
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| 24 |
Final Paper Presentations III |
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