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. I will try to prepare write-up for each lecture.
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.
write-up 3 (to be completed).
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. I will present the proof of FLT for exponent n = 3.
write-up 5 (to be completed).
This topic is not discussed in my lecture notes, so do not forget to come to
the lecture!
October 2025