| Abstract [eng] |
The topics examined in this thesis were the subject of my research as a PhD student at the Faculty of Mathematics and Informatics of Vilnius University. The presented investigation concerns the existence of composite numbers in some special sequences, such as the sequence of integer parts of powers of a fixed number and a linear recurrence sequence consisting of integer numbers. The thesis consists of the introduction, 3 sections, conclusions and bibliography. In Section 1 we consider composite numbers in the sequences of integer parts of powers of rational numbers and prove that the sequence [ξ(5/4)^n], n=1,2,..., where ξ is an arbitrary positive number, contains infinitely many composite numbers. Furthermore, it is shown that there are infinitely many positive integers n such that ([ξ(5/4)^n]; 6006)>1, where 6006 = 2•3•7•11•13. Similar results are obtained for shifted powers of some other rational numbers. In particular, the same is proved for the sets of integers nearest to ξ(5/3)^n and to ξ(7/5)^n, n=1,2,.... The corresponding sets of possible divisors are also described. In Section 2 we consider composite numbers in the binary linear recurrence sequences and prove that for every pair of integer numbers (a; b), where b≠0 and (a; b)≠(±2; -1), there exist two positive relatively prime composite integers x_1, x_2 such that the sequence given by x_{n+1}=ax_n+bx_{n-1}, n=2,3,..., consists of composite terms only, i.e., the absolute value of each term is a composite integer for each positive integer n. In Section 3 we consider the sets of the numbers expressible by some special linear form in connection with Egyptian fractions. We investigate which numbers belong to those sets. |
