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. 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. write-up 5 (to be completed).
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.
write-up 6 (to be completed).
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). write-up 7
(to be completed).
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!).
write-up 8 (to be completed).
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).
December 2025