| 1 |
Infinitude of The Primes
Formulas Producing Primes?
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| 2 |
Summing Powers of Integers, Bernoulli Polynomials |
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| 3 |
Generating Function for Bernoulli Polynomials
The Sine Product Formula and $\zeta(2n)$
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Assignment 1 due |
| 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 |
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| 6 |
Fermat's Little Theorem and Applications |
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| 7 |
Fermat's Great Theorem |
Assignment 2 due |
| 8 |
Applications of Fermat's Little Theorem to Cryptography: The RSA Algorithm |
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| 9 |
Averages of Arithmetic Functions |
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| 10 |
The Arithmetic-geometric Mean; Gauss' Theorem |
Assignment 3 due |
| 11 |
Wallis's Formula and Applications I |
Topic proposal and full outline of the paper due |
| 12 |
Wallis's Formula and Applications II (The Probability Integral)
Stirling's Formula
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| 13 |
Stirling's Formula (cont.) |
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| 14 |
Elementary Proof of The Prime Number Theorem I |
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| 15 |
Elementary Proof of The Prime Number Theorem II: Mertens' Theorem, Selberg's Formula, Erdos' Result |
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| 16 |
Short Analytic Proof of The Prime Number Theorem I (After D. J. Newman and D. Zagier) |
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| 17 |
Short Analytic Proof of The Prime Number Theorem II: The Connection between PNT and Riemann's Hypothesis
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First draft of paper due |
| 18 |
Discussion on the First Draft of the Papers and some Hints on how to Improve the Exposition and use of Latex |
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| 19 |
Euler's Proof of Infinitude of Primes
Density of Prime Numbers
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| 20 |
Definition and Elementary Properties of Fibonacci Numbers, Application to the Euclidean Algorithm
Binet's Formula
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Second draft of paper due |
| 21 |
Golden Ratio
Spira Mirabilus
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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 |
Final version paper due |