Download Pries: 405 Number Theory, Spring 2012. Project Topics. Combinatorial number theory:

Pries: 405 Number Theory, Spring 2012. Project Topics.
Combinatorial number theory:
1. Continued fractions: Stein 5.1, 5.2, 5.6.
2. Infinite continued fractions: Stein 5.3, 5.5.
3. Generating functions. Fibonacci numbers modulo p.
4. Binomial theorem and sums of powers.
5. Mersenne primes and perfect numbers.
6. Multiplicative functions.
Algebraic number theory:
1. Quadratic continued fractions and Pell’s equation: Stein 5.5
2. Sums of squares and factorization in Z[i].
3. Quadratic rings.
4. Fermat’s Last Theorem for exponent 4.
5. Every positive number is the sum of 4 squares.
Analytic number theory:
1. Prime number theorem
2. Riemann-Zeta function.
3. Diophantine approximation.
4. Transcendental numbers.
Geometric number theory:
1. Pythagorean triples and the unit circle.
Cryptography:
1. Primality testing: Stein 2.4.
2. Diffie Hellman key exchange and discrete log problem: Stein 3.1-3.2
3. Discrete log problem and El Gamal cryptosystem:
4. RSA cryptosystem and attacks: Stein 3.3 -3.4
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