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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 1