Algebraická teorie čísel / Algebraic Number Theory, should be more correctly and precisely: Algebraic Methods
in Number Theory and Combinatorics, NDMI066, fall/winter term 2025/26
As in the last year I will lecture on P. Mihailescu's proof of Catalan's conjecture and I will use the
booklet by R. Schoof, as well as the other two
books mentioned below.
Catalan's conjecture is nicely summarized by the very first page of
the book
by P. Ribenboim (which is from the era when Catalan's conjecture was
still a conjecture and not a theorem): CATALAN'S 3^2 - 2^3=1! X^U - Y^V=1? CONJECTURE Are 8 and 9 the Only Consecutive Powers?
The third
book
on Catalan's conjecture is due to Yu. F. Bilu, Y. Bugeaud and M. Mignotte. You see what an exciting and interesting
problem/theorem it is. I will try to prepare write-up for each lecture.
Lecture 1 on October 7, 2025. The equation x^4 - 3y^2 = 1 has only the trivial solution (+-1, 0). It follows
that the only solutions of the equation x^2 - y^3 = 1 are (+-3, 2), (+-1, 0) and (0, -1). This (elementary) resolution of the
special case of Catalan's equation is due to myself (in 1989): write-up 1 (to be completed).
Lecture 2 on October 14, 2025. V. Lebesgue's theorem (1850): for every odd m > 2, the equation x^m - y^2 = 1 has only the trivial
solution 1, 0. write-up 2 (complete).
Lecture 3 on October 21, 2025. Properties of the p-adic order. The domain Z[i] of Gaussian integers is
UFD. write-up 3 (complete).
October 28, 2025. State holiday
Lecture 4 on November 3, 2025. write-up 4 (to be completed).
November 2025