Addendum to the paper:"On two theorems of Gelfond and some of their applications" (Acta Arith. 13 (1967), pp. 177-236)
Almost perfect powers in arithmetic progression
Almost perfect powers in consecutive integers
Almost powers in the Lucas sequence
The famous problem of determining all perfect powers in the Fibonacci sequence and in the Lucas sequence has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations , with and , for all primes and indeed for all but primes . Here the strategy of [10] is not sufficient due to the sizes of...
Almost squares in arithmetic progression (II)
An theorem over function fields and applications
We provide a lower bound for the number of distinct zeros of a sum for two rational functions , in term of the degree of , which is sharp whenever have few distinct zeros and poles compared to their degree. This sharpens the “-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface contains only finitely many rational or elliptic curves,...
An application of a lower bound for linear forms in two logarithms to the Terai-Jeśmanowicz conjecture
An extension of a theorem of Euler
An upper bound for the G.C.D. of two linear recurring sequences
Application of the explicit abc-conjecture to two Diophantine equations
Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations
Arithmetic progressions with common difference divisible by small primes
Arithmetical study of recurrence sequences
Baker's explicit abc-conjecture and applications
Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)
This is an exposition of the recent work of Bugeaud, Hanrot and Mihăilescu showing that Catalan’s conjecture can be proved without using logarithmic forms and electronic computations.
Catalanova domněnka dokázána
Catalan’s conjecture
The subject of the talk is the recent work of Mihăilescu, who proved that the equation has no solutions in non-zero integers and odd primes . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebratedconjecture of Catalan (1843): the only solution to in integers and is . Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute...
Catalan's equation has no new solution with either exponent less than 10651.
Common divisors of and over function fields.
Compact integers and factorials