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.
Exam questions. 1. Prove that x^4 - 3y^2 = 1 has no nonzero solution (Chapter 1.3 of Lecture Notes).
2. Prove that the only solutions of x^2 - y^3 = 1 are (+-3,2), (+-1,0) and (0,-1) (Chapter 1.3 of Lecture Notes).
3. Prove that x^m - y^2 = 1 for odd m > 1 has no nonzero solution (Chapter 2.2 of Lecture Notes).
4. Prove that x^2 - y^q = 1 for any prime q > 3 has no nonzero solution (Chapter 3.2 of Lecture Notes).
5. Prove that if x^p - y^q = 1 for some primes p > q > 2 and nonzero integers x and y, then q divides x
(Chapter 4.2 of Lecture Notes).
6. Prove that if x^p - y^q = 1 for some primes p > q > 2 and nonzero integers x and y, then p divides y
(Chapter 4.3 of Lecture Notes).
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). See lecture notes (Chapter 1).
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. See lecture notes (Chapter 2).
Lecture 3 on October 21, 2025. Properties of the p-adic order. The domain Z[i] of Gaussian integers is
UFD. See lecture notes (Chapter 2).
October 28, 2025. State holiday
Lecture 4 on November 4, 2025. Theorem: every Euclidean domain is UFD, proof. Beginning of the proof
of Theorem (Chao Ko, 1965): For every prime q > 3, the equation x^2 - y^q = 1 has no solution in nonzero integers x, y.
See lecture notes (Chapter 3).
Lecture 5 on November 11, 2025. Conclusion of the proof of Chao Ko's theorem. See
lecture notes (Chapter 3).
Lecture 6 on November 18, 2025. Cassels's relations - part 1. See lecture notes (Chapter 4).
Lecture 7 on November 25, 2025. Cassels's relations - part 2. See lecture notes (Chapter 4).
Lecture 8 on December 2, 2025. The obstruction group - part 1. See lecture notes (to be written).
Lecture 9 on December 9, 2025. The obstruction group - part 2. See lecture notes (to be written).
Lecture 10 on December 16, 2025. Euler's theorem: the only rational solutions of x^2 - y^3 = 1 are (+-3, 2), (+-1,0)
and (0,-1). See lecture notes (Chapter 1).
December 2025