Algebraická teorie čísel / Algebraic Number Theory, should be more correctly and precisely: Algebraic Methods
in Number Theory and Combinatorics, NDMI066, winter term 2024/25
Friday 17:20-18:50 in S6 (on November 29 elsewhere)
This time 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 restarted my Diophantine Tetralogy project and begin to write its first
part (state by November 1, 2024).
In it I will cover, besides other things, my lectures in the course, with the exception of lecture 1.
Lecture 1 on October 4, 2024. lecture 1 (more or less). The Catalan problem and
a very rough outline of Mihailescu's proof.
Lecture 2 on October 11, 2024. Chapter 2 of my text The equation x^m - y^2 = 1 for odd
m > 1 has no nonzero solution - the theorem of V. Lebesgue (1850).
Lecture 3 on October 18, 2024. Chapters 3.1 and 3.2 of my text. My 1989 solution of the equation
x^2 - y^3 = 1, all solutions are (+-3, 2), (+-1, 0) and (0, -1).
Lecture 4 on October 25, 2024. Again Chapters 3.1 and 3.2 of my text. My 1989 solution of the equation
x^2 - y^3 = 1.
Lecture 5 on November 1, 2024. Chapter C.1 of my text. UFDs, Euclidean domains and
a proof of the theorem that Euclidean => UFD.
Lecture 6 on November 8, 2024. Chapter 4.1 of my text. Solving the Diophantine equation x^2 - y^q = 1,
q > 3 is a prime number - the theorem of Chao Ko (1965) and its proof by Chein (1976).
Lecture 7 on November 15, 2024. Chapter 5 of my text (more or less complete). The relations of Cassels
(1953, 1960): if x, y is a nonzero solution of x^p - y^q = 1, where p, q > 2 are primes, then p divides y and q divides x.
Lecture 8 on November 22, 2024. Conclusion of the proof of the harder relation of Cassels. Proof of the lemma on binom(a/q, k).
Lecture 9 on November 29, 2024. On number fields (mostly after Appendix A of the BBM book). See Appendix D.1 (to be written)
of
my text
Exam questions (to be updated): 1. ...
November 2024