Úvod do teorie čísel, NMAI040, ZS 2022/23
Na místu a čase konání přednášek se ješté dohodneme. Budu postupovat podle předloňských přednášek.
Exam is oral, with written preparation
Exam questions: 1a. Dirichlet's theorem on Diophantine approximation and its applications. 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 aritmetical proof. 3a. Lattices and their properties, the prof of the theorem on Farey fractions by means of lattices. 3b. The geometric proof of Lagrange's four-squares theorem. 4a. Prove Čebyšev's bounds on the prime counting function pi(x). 4c. Give 5 (five) proofs of the infinitude of prime numbers (so look for one more proof in addition to the LN). 5. Explain the theory of quadratic residues including the reciprocity law. 6. State and prove the pentagonal identity for integer partitions.
October 2022