## Introduction to Number Theory, NMAI040, winter term 2023/24

Friday 15:40-17:10 in S5. See what was three years ago. Here are my lecture
notes (LN) from 2006.

Oct 6, 2023, lecture 1 (partially updated). **1. Diophantine approximation.**
Dirichlet's theorem (on Diophantine approximation). Farey fractions. Hurwitz's theorem (an ultimate strengthening of Dirichlet's theorem).

Oct 13, 2023, lecture 2 (to be updated). Liouville's theorem (existence of transcendental numbers). The Hermite-Hilbert theorem
(transcendence of e).

Oct 20, 2023, lecture 3 (to be updated). **2. Diophantine equations.** Remarks without proofs: Skolem's reduction,
Siegel's theorem (solvability of quadratic D. equations is decidable ...), DPRM Theorem (... but it is not for higher degrees). The Pell equation(s). Lagrange's
theorem (every Pell equation has infinitely many solutions)

Oct 27, 2023, lecture 4 (to be updated). A theorem on the structure of solutions of the Pell equation. Pythagorean triples and
a theorem on them.

Nov 3, 2023, lecture 5 (to be updated). FLT for n=4. Partially FLT for n=3, Eulers proof.

Nov 10, 2023, lecture 6 (to be updated). **3. Geometry of numbers.** Lattices and their volumes. Geometric proof of
Haros's theorem on Farey fractions. Minkowski's theorem on convex body.

Nov 17, 2023, no lecture: state holiday.

Nov 24, 2023, lecture 7 (to be updated). Lagranges theorem on four squares via Minkowski's theorem.
**4. Prime numbers.** Gauss' circle problem.

Dec 1, 2023, lecture 8 (to be updated). Dirichlet's divisor problem. The infinitude of primes, two proofs.

Dec 8, 2023, lecture 9 (to be updated). Two more proofs of the infinitude of primes. Starting the proof of
P. Erdos of Chebyshev's estimates of the prime number counting function.

Dec 15, 2023, lecture 10. Completing the proof of P. Erdos of Chebyshev's estimates of the prime number counting function. **5. The quadratic
reciprocity law.** Basic definitions around quadratic (non-)residues. Sections 4.2 and 5.1 of the LN.

Dec 22, 2023, teaching was cancelled because of the
sad events on FF UK the previous day.

Exam is oral, with written preparation
**Exam questions (updated!):** **1a.** Dirichlet's theorem on Diophantine approximation and
its application(s). **1b.** Existence of transcendental numbers: Liouville's inequality. **1c.** The proof of transcendence of the number e.
**2a.** Describe the theory of Pell (Diophantine) equations. **2b.** Lagrange's four-squares theorem, the geometric proof. **3a.**
Lattices and their properties, the prof of the theorem on Farey fractions by means of lattices. **4a.** Prove Chebyshev's bounds on the prime
counting function pi(x). **4c.** Give 5 (five) proofs of the
infinitude of prime numbers (possibly look for them in the LN). **5.** Explain the theory of quadratic residues including
the reciprocity law. **6.** State and prove Euler's odd-distinct identity for integer partitions.

January 2024