Introduction to Number Theory, NMAI040, winter term 2024/25
Friday 15:40-17:10 in S6 (on November 29 elsewhere) I will be teaching similarly to my courses in previous years and will follow
my lecture notes from 2006. Last year I attempted to work our lecture notes
for each lecture, and I failed. I will attempt doing it this year too. And I am failing too, since I devote most of my time and effort to the
other course Algebraic number theory.
lecture 1 (preliminary) on October 4, 2024. 1. Diophantine approximation.
Theorems of Dirichlet and Hurwitz on Diophantine approximation. Farey fractions. The Lonely Runner Conjecture.
lecture 2 (preliminary) on October 11, 2024. Conclusion of the proof of the Hurwitz theorem.
Liouville's inequality and its application in existence of trascendental numbers.
lecture 3 on October 18, 2024. The Hermite-Hilbert theorem: the number e is trascendental.
lecture 4 on October 25, 2024. 2. Diophantine equations. The Pell equation. Lagrange (1770): every Pell equation
has a nontrivial solution. Every generalized Pell equation x^2 - dy^2 = m with nonzero m has either no solution or infinitely many.
lecture 5 on November 1, 2024. Proof that the group of positive solutions x + y*d^{1/2} of the Pell equation is isomorphic
to (Z, 0, +). Two derivations of the parametrization of Pythagorean triples (nonnegative coprime integers x, y, z such that x^2 + y^2 = z^2).
The theorem of de Fermat that x^4 + y^4 = z^2 has no nontrivial (xyz nonzero) solution.
lecture 6 on November 8, 2024. 3. Geometry of numbers. Lattices L in R^n. Fudamental parallelepipeds of L and vol(L).
Minkowski's theorem about a convex body B and a lattice L in R^n; proof. Lemma: if m in N is square-free then for some integers a, b the
number a^2 + b^2 + 1 is divisible by m; proof by a pigeonhole principle and the Chinese remainder theorem. The four-squares theorem
(Lagrange, 1770): every nonnegative integer is a sum of four (integral) squares; first half of the proof by Minkowski's theorem, we
defined B and L and complete the proof next time.
lecture 7 on November 15, 2024. Completion of the proof of Lagrange's theorem on four squares. The divisor problem and the theorem
of P. Dirichlet: sum_{n at most x}tau(n) = x*log x + (2gamma - 1)*x + O(x^{1/2}), where tau(n) is the number of divisors of n and gamma = 0.577...
is the Euler constant. Remarks on the history of the divisor problem. 4. Primes. Euclid's proof of infinitude of primes.
lecture 8 on November 22, 2024. Goldbach's proof of infinitude of primes. One of its forms uses Fermat's numbers F_n = 2^{2^n} + 1.
Remarks on the Gauss-Wantzel theorem: the regular plane n-gon can be constructed by ruler and compass <=> n = 2^m*product of different Fermat's
primes. P. Wantzel proved => ; see this article on
his proof. Fermat's primes known to date are F_0 = 3, F_1 = 5, F_2 = 17 (C. F. Gauss discovered an Euclidean construction of the
plane resular 17-gon), F_3 = 257 and F_4 = 65537, it is not known if there are any other. L. Euler showed that F_5 = 2^{32} + 1 is divisible
by 641. The proof of infinitude of primes due to Cass and Wildenberg (combinatorial simplification of Furstenberg's topological proof). The theorem of
P. L. Chebyshev: there are constants 0 < c_1 < c_2 such that for every x at least 2 we have c_1*x / log x <
pi(x) = the number of primes p at most x < c_2*x / log x. I finish the proof (due to P. Erdos) next time.
lecture 9 on November 29, 2024. Conclusion of the proof of Chebyshev's bounds. Mersenne primes: prime numbers of the form M_p = 2^n - 1 =
2^p - 1. The Lucas-Lehmer test for them (mentioned without proof). 5. Congruences. Quadratic (non-)residues modulo p. Legendres
symbol (a/p). Eulers criterion: (a/p) is a^{(p - 1)/2} modulo p (p>2). The quadratic reciprocity law and its two supplements.
lecture 10 on December 6, 2024. The Gauss lemma on quadratic residues. Combinatorial reciprocity lemma (CRL). Proof of the quadratic
reciprocity law.
lecture 11 on December 13, 2024. Proof of the CRL. 6. Integer partitions. Partitions and compositions of a natural number.
GFs for the sequences of numbers of partitions and compositions. Euler's simple identity (ESI): the # of partitions of n in odd parts =
the # of partitions of n in distinct parts. A proof by GFs.
lecture 12 on December 20, 2024. A bijective proof of ESI. The Cohen--Remmel theorem (CRT): If (A_n) and (B_n)
are two sequences of partitions such that for every finite set of indices I one has that ||U_{j in I}A_j|| = ||U_{j in I}B_j||, then
for every n, the # of partitions of n avoiding every A_j = the # of partitions of n avoiding every B_j; proof by PIE. Some
applications/corollaries of the CRT.
lecture 13 on January 10, 2025.
Exam is oral, with written preparation
Exam questions (final form): 1a. Dirichlet's theorem on Diophantine approximation and
its application(s). 1b. The Hurwitz theorem. 1c. Existence of transcendental numbers: Liouville's inequality. 1d. The
proof of transcendence of the number e.
2a. Describe the theory of Pell (Diophantine) equations. 2b. Fermat's theorem: x^4 + y^4 = z^2 has no nontrivial solution.
3a. Lagrange's four-squares theorem, the geometric proof. 4a. Prove Chebyshev's bounds on the prime counting function pi(x).
4c. Give several proofs of the infinitude of prime numbers (possibly look for them in the LN or elsewhere). 5. Explain the theory of
quadratic residues including the reciprocity law. 6a. State and prove Euler's odd-distinct parts identity, by GFs and bijectively.
6b. State and prove the Cohen--Remmel theorem on identities for integer partitions, and give some corollaries of it.
December 2024