Seminario: "On 1’s and other patterns in continued fraction expansions of square roots of prime numbers and their doubles" - Piotr Miska

Giovedì 29 giugno 2023 - ore 15:00
Aula Buzano - Dipartimento di Scienze Matematiche
Politecnico di Torino

"On 1’s and other patterns in continued fraction expansions of square roots of prime numbers and their doubles"
Piotr Miska
Jagiellonian University

Abstract: The aim of the talk is to present results concerning the appearance of consecutive 1’s in expansions of square roots of prime numbers into continued fractions. I will show three different approaches:
- by considering a polynomial of two variables such that all its values at positive integer arguments have square roots with periodic parts of continued fractions starting with three 1's,
- with assumption of Hypothesis H of Schinzel and Sierpiński,
- with the use of equidistribution modulo 1 of the sequence of square roots of consecutive prime numbers and the facts from the measure theory of continued fractions.
Next, I will focus on the problem of appearance of an odd number of times of a given number a or at least one 1 in continued fraction expansion of square roots of prime numbers and 2 times prime numbers.