Introduction to Number Theory, NMAI040, fall/winter term 2025/26
I will be teaching similarly to my courses in previous years and will follow
my lecture notes from 2006.
Exam questions. 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 (proof only part 2). 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. Lagrange's four-squares theorem, arithmetical proof. 3a. Lagrange's four-squares theorem, geometric proof.
4a. Prove Chebyshev's bounds on the prime counting function pi(x). 4b Derive asymptotics for the sum_{n < x}tau(n), where
tau(n) is the number of divisors of n. 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. TBA.
lecture 1 on September 30, 2025. 1. Diophantine approximations.
Theorems of Dirichlet and Hurwitz on Diophantine approximation. Farey fractions. The Lonely Runner Conjecture.
write-up 1 (complete).
lecture 2 on October 7, 2025. Examples of irrational numbers a, b > 0 such that the power a^b is
rational. Liouville's inequality: irrational algebraic numbers do not have very good rational
approximations. Thus 0.11000100000000000000000100... is a transcendental (non-algebraic) number.
Hilbert's proof of Hermite's theorem that the Euler number e = 2.71828... is transcendental -
we finish it next time. write-up 2 (complete).
lecture 3 on October 14, 2025. Conclusion of Hilbert's proof. 2. Diophantine equations.
Three famous exproblems: (1) The DPRM theorem, (2) FLT and (3) The Catalan problem. Pell equation:
if it has a nontrivial solution, then it has infinitely many solution.
lecture 4 on October 21, 2025. Lagrange's theorem (1770): every Pell equation has a nontrivial solution.
Corollary: if x^2 - dy^2 = m, where d > 0 is a non-sqquare and m is nonzero, has a solution, then
it has infinitely many solutions. Theorem (group structure): if d is as before and M_d =
{a + b*d^{1/2} > 0: a^2 - db^2 = 1}, then (M_d, 1, *), where * is the standard multiplication of real
numbers, is a group isomorphic to the infinite cyclic group (Z, 0, +).
write-up 4 (complete).
October 28, 2025. State holiday
lecture 5 on November 4, 2025. In the write-up I will present the proof of FLT for exponent n = 3.
In the lecture I started Chapter 3 on geometry of numbers. n-dimensional lattices and their volumes.
Minkowski's theorem on convex body. Theorem (Lagrange, 1770): every nonnegative integer is a sum of four
squares. Beginning of the proof.
lecture 6 on November 11, 2025. Conclusion of the proof of Lagrange's theorem. An arithmetic proof
of Lagrange's theorem based on Euler's four-square identity.
lecture 7 on November 18, 2025. Two asymptotics of summatory functions of arithmetic functions:
sum_{x < n}r_2(n) = pi*x + O(x^{1/2}) (the circle problem of Gauss) and sum_{x < n}tau(n) =
x*log(x) + (2gamma - 1)x + O(x^{1/2}) (the divisor problem of Diriclet).
lecture 8 on November 25, 2025. Chapter 4: Prime numbers. Five proofs of infinitude of the
set of primes. P1 (Euclid). P2 (Goldbach): by means of the Fermat numbers F_n = 2^{2^n} + 1, comments on them.
P3 (Erdos): gives the bound pi(x) > c*log x. P4 (Cass and Wildenberg, 2003): by means of periodic sets of integers.
P5 (Euler): prod_{i=1}^k 1/(1 -1/p_i) >= 1 + 1/2 + 1/3 + ... . Theorem (Čebyšev, cca 1850): cx/(log x)
< pi(x) < dx/(log x), beginning of the proof. Next time we resume by proving Legendre's formula for ord_p(n!).
lecture 9 on December 2, 2025. Proof of Legendre's formula and proof of both Čebyšev's bounds. Some more
prime number asymptotics (no proofs): three formulas of Mertens and the theorem of Hardy and Ramanujan on the
normal order of omega(n) and Omega(n).
lecture 10 on December 9, 2025.
Chapter 5: The quadratic reciprocity law. Properties of
Legendre's symbol (a/p). Euler's citerion and Gauss's lemma. Two supplement to the reciprocity law.
lecture 11 on December 16, 2025. Two lemmas and the proof of the quadratic reciprocity law. Chapter 6:
Integer partiions. Euler's odd vs. distinct parts identity, proof by generating functions.
December 2025