Course description:

According to Gauss, number theory is the Queen of Mathematics. The course tries to help you to creep into the Queen's favor. There are no recipes, but assiduous courting and devoted service will certainly bear fruits. The course offers an introduction to a discipline rich in interesting solved and unsolved problems, some dating back to very ancient times. We start from the very first steps to conclude with an outlook to certain aspects of advanced number theory. Most notions and proofs do not require too many technical details, and our approach is generally through ``why'' instead of ``how''. Problem solving forms an essential and integrated part of the course. Among others we plan to consider the following questions:

• Which positive integers can be written as the sum of two squares?
• How can you make $150000 by finding a prime number?
• Is 2^k+1 a prime if k is (a) 101; (b) 2022; (c) 16; (d) 32; (e) 2³³; (f) 2^182332954 ?
• How does an innocent-looking 2000 years old question lead — in the view of Erdôs — to the hardest though not the most important problem of mankind?

Topics covered:

• Divisibility: Euclidean algorithm, greatest common divisor, unique factorization.
• Primes: Mersenne and Fermat primes, some famous conjectures, primes in arithmetic progressions, primality testing and cryptography.
• Congruences: Euler's function φ(n), Euler-Fermat theorem, linear and quadratic congruences, Chinese Remainder Theorem, order, primitive roots, Legendre and Jacobi symbols and their applications.
• Arithmetic functions: multiplicativity, number and sum of divisors ( d(n) and σ(n) ), perfect numbers.
• Diophantine equations: linear equation, Pythagorean triples, Fermat's Last Theorem, sums of squares, some typical methods for solving Diophantine equations.