As I am lecturing in English this term, information here will be in English too. For this course in previous years see here (in Czech, though). Syllabus
and annotation in SIS. Here are lectures notes in English, here are some errata.

Lectures are on Monday in S5 (2nd floor of the Malostranske square building) in 10:40-12:10.

I built this course on the classical book G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers and (almost) everything I am lecturing about can be found there. In the lectures I may provide further references and literature.

Examination is in oral form (you get a question from the following list, prepare for some time and then explain it to me). Terms of examinations: see SIS (or by appoinment). Questions (now updated for 2019): 1a. Dirichlet's theorem on diophantine approximations and applications. 1b. Existence of transcendental numbers by Liouville's inequality. 1c. Proof that e is transcendental. 2a. Theory of the Pell equation. 2b. The Stothers-Mason theorem and proof of the FLT for polynomials. 3a. Lattices and their properties, Farey fractions via plane lattices. 3b. Geometrical proof of Lagrange's 4 squares theorem. 4a Give five proofs of infinitude of prime numbers. 4b. Čebyšev's estimates of the primes counting functione pi(x). 5. Prove the quadratic reciprocity law. 6a. Prove by PIE the odd-parts-versus-distinct-parts partition identity. 6b. Prove Euler's pentagonal identity.

Lectures are on Monday in S5 (2nd floor of the Malostranske square building) in 10:40-12:10.

I built this course on the classical book G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers and (almost) everything I am lecturing about can be found there. In the lectures I may provide further references and literature.

Examination is in oral form (you get a question from the following list, prepare for some time and then explain it to me). Terms of examinations: see SIS (or by appoinment). Questions (now updated for 2019): 1a. Dirichlet's theorem on diophantine approximations and applications. 1b. Existence of transcendental numbers by Liouville's inequality. 1c. Proof that e is transcendental. 2a. Theory of the Pell equation. 2b. The Stothers-Mason theorem and proof of the FLT for polynomials. 3a. Lattices and their properties, Farey fractions via plane lattices. 3b. Geometrical proof of Lagrange's 4 squares theorem. 4a Give five proofs of infinitude of prime numbers. 4b. Čebyšev's estimates of the primes counting functione pi(x). 5. Prove the quadratic reciprocity law. 6a. Prove by PIE the odd-parts-versus-distinct-parts partition identity. 6b. Prove Euler's pentagonal identity.

lecture 1, October 1, 2018. 1. Diophantine approximation. Dirichlet's
theorem (DT): part 1 says that |a - p/q| < or = 1/qQ has for any
real number a and an integer Q > 1 at least one rational solution
p/q
with 0 < q < Q, part 2 says that |a - p/q| < q^{-2}has for any
irrational real number a infinitely many rational solutions p/q. As a corollary of part 1
of DT I proved the Euler-Fermat theorem: every prime number of the form p =
1+4n is a sum of two squares. We used a Lemma: -1 is quadratic residue
modulo such p. Proof: by partitioning
Z_p^x. Hurwitz's theorem stated. The Farey fractions, their main property will be proven next time.

lecture 3, October 15, 2018. Conclusion of the proof of J. Liouville's inequality. Liouville's numbers: a is a L. number, if for every n there is a fraction p/q different from a with q>1 s. t. | a- p/q | < 1/q^n. Every L. number is transcendental, but neither pi nor e are L. numbers. Theorem (Hermite, 1873): e is transcendental, proof (after D. Hilbert, 1890). Remarks on 1) Roth's theorem, 2) Hilbert's 7th problem and 3) an open problem: prove that gamma is irrational.

lecture 4, October 22, 2018. 2. Diophantine equations. Hilbert's
10th problem and it's (non-)solution by Matijasevič, Davis, Putnam and
Robinson. Pell equation P(d) is x^2 - dy^2 = 1 where d > 0 is an
integer and is not a square. Let S = {x + yd^{1/2} | x, y is an
integral solution of P(d)} and S^+ = {a in S | a > 0}. Theorem: 1)
(S^+, *) (* is standard multiplication of real numbers) is isomorphic
to (Z, +) and thus is (the) infinite cyclic group; hence, trivially, 2)
(S, *) is isomorphic to (Z, +) times (Z_2, +), proof. Corollary: every
generalized P(d), x^2 - dy^2 = m where d is as before and m is a
nonzero integer, has either no solution x, y in Z or infinitely many
(for m = 0 it has just one solution 0, 0), proof. lecture 2, October 8, 2018. Proof
of the Farey-Cauchy theorem on F. fractions. Proof, by F. fractions, of
Hurwitz's theorem: (1) for any irrational real a there are infinitely
many fractions p/q s. t. | a - p/q | < 1/5^{1/2}q^2 and (2) if
b=(5^{1/2} - 1)/2 and A > 5^{1/2} then | b - p/q | < 1/Aq^2 for
only finitely many fractions p/q. J. Liouville's inequality (1844): if
a is an algebraic real number with deg(a) = n then for some constant c
> 0 and every fraction p/q different from a one has | a - p/q
| > c/q^n; conclusion of the proof next time. Corollary: for every
integer k > 1 the number (sum of infinite series for n = 1, 2, ...)
sum_n k^{-n!} is transcendental.

lecture 3, October 15, 2018. Conclusion of the proof of J. Liouville's inequality. Liouville's numbers: a is a L. number, if for every n there is a fraction p/q different from a with q>1 s. t. | a- p/q | < 1/q^n. Every L. number is transcendental, but neither pi nor e are L. numbers. Theorem (Hermite, 1873): e is transcendental, proof (after D. Hilbert, 1890). Remarks on 1) Roth's theorem, 2) Hilbert's 7th problem and 3) an open problem: prove that gamma is irrational.

lecture 5, October 29, 2018. Remarks on Thue equations and on A. Baker's effective bound on solutions x, y of x^3 - 2y^3 = m (I promised to tell it next time). FLT (Fermat's Last Theorem): if x^n + y^n = z^n where x, y, z, n > 0 are integers, then n < 3. Case n=2: description of all solutions, so called Pythagorean triples, proof. Case n=4: no solution x, y, z exists, proof. FLT for polynomials: if x^n + y^n = z^n where n > 0 is an integer and x, y, z are univariate polynomials in C[t], not all constant and coprime, then n < 3. Deduction from the Stothers-Mason theorem which will be proven next time. I will also give a direct proof of the FLT for polynomials.

lecture 6, November 5, 2018. Information on A. Baker's effective bound (1964): if x, y, c are integers such that x^3 - 2y^3 = c then |x|, |y| < (300000|c|)^{23}. A direct proof of the FLT for polynomials by unique factorization in C[t] and by infinite descend. A proof of the Stothers-Mason theorem by (formal) logarithmic derivatives of rational functions. Remarks on the ABC conjecture and the 2012 claimed Mochizuki's proof. 3. Geometry of numbers. A lattice in R^m, its basis, and its fundamental parallelepiped. The volumes of f. p-s do not depend on the choice of basis, proof next time.

lecture 7, November 12, 2018. Proof that the volumes of f. p-s do not depend on the choice of basis. Proof of the Cauchy-Farey thm. on F. fractions by the lattice Z^2. Lemma: a^2 + b^2 + 1 = 0 modulo m has a solution for any square-free number m, proof. Minkowski's theorem on convex body, proof next time. Deduction of Lagrange's four-squares theorem from Minkowski's theorem.

lecture 8, November 19, 2018. Proof of Minkowski's theorem. 4. Prime numbers. FTA: every integer n > 0 is a unique product of primes. Euclid's theorem: there are infinitely many primes. Ten proofs of Euclid's theorem. 1. Euclid's proof: if p_1, ..., p_k are primes then 1 + p_1p_2...p_k is divisible by a prime different from all p_i. 2. Goldbach: 2^{2^n} + 1, n = 0, 1, ..., are pairwise coprime. 3. Erdos (1): consider the mapping n |-> (A, b^2) where A is the set of all primes that have in the factorization of n odd exponents and b^2 = n / the product of the primes in A. 4. Erdos (2): 2^n <= binom(2n, n) <= (2n)^{pi(2n)}. 5. Furstenberg (1955), simplified and formulated in the right way by Cass and Wildenberg (2003): Z \ {-1, 1} = U_{primes p}(pZ), the left side is a non-periodic subset of Z but the right side is, if there are only finitely many primes, periodic - conclusion of this proof next time.

lecture 9, November 26, 2018. Conclusion of proof 5. 6. A combinatorial proof: For any k fixed primes p_1, ..., p_k one has |{n < x | n=p_1^{a_1}...p_k{a_k}}| = O((1+log n)^k). 7. Euler: prod_p(1 - p^{-1})^{-1} = 1 + 1/2 + 1/3 + ... = oo. 8. Sylvester: prod_{p<x}(1 - p^{-1})^{-1} > 1 + 1/2 + 1/3 + ... 1/([x] - 1). 9. Euler: for every s > 1, prod_p(1 - p^{-s})^{-1} = 1 + 1/2^s + 1/3^s + ... < oo. 10. Formal proof: formally (in the ring of formal Dirichlet series) (1 + 1/2^s + 1/3^s + ...)(1 + mu(2)/2^s + mu(3)/3^s + ... ) = 1 where mu(n) is the Moebius function.Theorem (Cebysev, 1850): cx/log x < pi(x) < dx/log x, proof - it has remained to prove just the upper bound which is based on the bound sum_{p<x}log p < cx.

lecture 10, December 3, 2018. 5. Quadratic residues and non-residues. That was the plan but nobody showed up.

lecture 11, December 10, 2018. Quadratic
residues and non-residues modulo p. For p > 2 their numbers are the
same, equal to (p - 1)/2. Legendre's symbol (a/p) and its properties:
1) depends only on the residue class of a mod p, 2) (Euler's
criterion): (a/p) =(mod p) a^{(p-1)/2} and 3) multiplicativity (ab/p) =
(a/p)(b/p), proof. The 1st supplement (to the reciprocity law): (-1/p)
= 1 iff p is 1 mod 4. Gauss' lemma: (a/p) = (-1)^{m(a)} = (-1)^{n(a)}
where m(a), resp. n(a), = the number of numbers in {(p+1)/2, ..., p-1},
resp. in {-(p-1)/2, ..., -1}, congruent mod p to some ka for k in {1,
2, ..., (p-1)/2}, proof. Corollary: (the 2nd supplement) (2/p) = 1 iff
p is 1 or 7 mod 8, proof. The reciprocity law: (p/q) = (q/p), where p,
q > 2 are distinct primes, if and only if p or q is 1mod 4
(i.e., (p/q) = -(q/p) iff both p and p are 3 mod 4), proof next time.

lecture 12, December 17, 2018. Proof of the quadratic reciprocity law, by means of two lemmas. L1: S(a, b) + S(b,a) = (a - 1)(b - 1)/4, where S(a, b) = sum_{i=1}^{(a-1)/2}[ib/a] and a, b> 1 are coprime odd numbers, proof. L2: (a/p) = (-1)^{S(p, a)} where p>2 is a prime and a>1 is an odd number not divisible by p, proof. 6. Integer partitions. Basic terminology. Two identities of L. Euler: the pentagonal identity and the odd-parts-versus-distinct-parts identity, proofs next time.

last lecture 13, January 7, 2019. The odd-parts-versus-distinct-parts identity, (i) proof by generating functions and (ii) proof by PIE. Euler's pentagonal identity, Franklin's bijective proof.

January 2019