Temos, nagrinėjamos šioje disertacijoje, buvo doktorantūros studijų Vilniaus universiteto Matematikos ir informatikos fakultete objektas. Pateikti tyrimai yra susiję su sudėtinių skaičių egzistavimu tokiose sekose kaip fiksuoto skaičiaus laipsnių sveikųjų dalių seka bei tiesinė rekurentinė seka, sudaryta iš sveikųjų skaičių. Disertaciją sudaro įvadas, 3 skyriai, išvados ir literatūros sąrašas. Pirmame skyriuje nagrinėjami sudėtiniai skaičiai racionaliųjų skaičių laipsnių sveikųjų dalių sekoje bei yra įrodoma, kad sekoje [ξ(5/4)^n], n=1,2,..., kur ξ yra bet koks teigiamas skaičius, yra be galo daug sudėtinių skaičių. Be to, įrodoma, kad yra be galo daug tokių natūraliųjų skaičių n, kad ([ξ(5/4)^n]; 6006)>1, čia 6006 = 2•3•7•11•13. Įrodoma panašių rezultatų pastumtoms kai kurių kitų racionaliųjų skaičių sekoms. Pavyzdžiui, tas pats įrodoma sveikųjų skaičių, esančių arčiausiai ξ(5/3)^n bei ξ(7/5)^n, n=1,2,..., sekoms. Vėlgi nurodomos atitinkamos galimų daliklių aibės. Antrame skyriuje nagrinėjami sudėtiniai skaičiai antros eilės tiesinėse rekurentinėse sekose bei įrodoma, kad kiekvienai tokiai sveikųjų skaičių porai (a; b), kad b≠0 ir (a; b)≠(±2; -1), egzistuoja tokie du natūralieji tarpusavyje pirminiai skaičiai x_1, x_2, kad sekoje, apibrėžtoje lygtimi x_{n+1}=ax_n+bx_{n-1}, n=2,3,..., visų narių moduliai yra sudėtiniai skaičiai. Trečiame skyriuje egiptietiškų trupmenų kontekste nagrinėjamos skaičių, užrašomų tam tikru tiesiniu pavidalu, aibės. Ieškoma, kokie skaičiai... [toliau žr. visą tekstą] / 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... [to full text]

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... [to full text] / Temos, nagrinėjamos šioje disertacijoje, buvo doktorantūros studijų Vilniaus universiteto Matematikos ir informatikos fakultete objektas. Pateikti tyrimai yra susiję su sudėtinių skaičių egzistavimu tokiose sekose kaip fiksuoto skaičiaus laipsnių sveikųjų dalių seka bei tiesinė rekurentinė seka, sudaryta iš sveikųjų skaičių. Disertaciją sudaro įvadas, 3 skyriai, išvados ir literatūros sąrašas. Pirmame skyriuje nagrinėjami sudėtiniai skaičiai racionaliųjų skaičių laipsnių sveikųjų dalių sekoje bei yra įrodoma, kad sekoje [ξ(5/4)^n], n=1,2,..., kur ξ yra bet koks teigiamas skaičius, yra be galo daug sudėtinių skaičių. Be to, įrodoma, kad yra be galo daug tokių natūraliųjų skaičių n, kad ([ξ(5/4)^n]; 6006)>1, čia 6006 = 2•3•7•11•13. Įrodoma panašių rezultatų pastumtoms kai kurių kitų racionaliųjų skaičių sekoms. Pavyzdžiui, tas pats įrodoma sveikųjų skaičių, esančių arčiausiai ξ(5/3)^n bei ξ(7/5)^n, n=1,2,..., sekoms. Vėlgi nurodomos atitinkamos galimų daliklių aibės. Antrame skyriuje nagrinėjami sudėtiniai skaičiai antros eilės tiesinėse rekurentinėse sekose bei įrodoma, kad kiekvienai tokiai sveikųjų skaičių porai (a; b), kad b≠0 ir (a; b)≠(±2; -1), egzistuoja tokie du natūralieji tarpusavyje pirminiai skaičiai x_1, x_2, kad sekoje, apibrėžtoje lygtimi x_{n+1}=ax_n+bx_{n-1}, n=2,3,..., visų narių moduliai yra sudėtiniai skaičiai. Trečiame skyriuje egiptietiškų trupmenų kontekste nagrinėjamos skaičių, užrašomų tam tikru tiesiniu pavidalu, aibės. Ieškoma, kokie skaičiai... [toliau žr. visą tekstą]

In 1878, Lucas \cite{lucas} investigated the sequences $(\ell_n)_{n=0}^\infty$ where $$\ell_n=\frac{\alpha^n-\beta^n}{\alpha-\beta},$$ $\alpha \beta$ and $\alpha+\beta$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. Lucas sequences are divisibility sequences; if $m|n$, then $\ell_m|\ell_n$, and more generally, $\gcd(\ell_m,\ell_n)=\ell_{\gcd(m,n)}$ for all positive integers $m$ and $n$. Matijasevic utilised this divisibility property of Lucas sequences in order to resolve Hilbert's 10th problem. \noindent In 1930, Lehmer \cite{lehmer} introduced the sequences $(u_n)_{n=0}^\infty$ where \begin{eqnarray*} u_n& = & \frac{\alpha^{n}-\beta^n}{\alpha^{\epsilon(n)}-\beta^{\epsilon(n)}},\\ \epsilon(n)&=&\left\{\begin{array}{ll} 1, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 1 \pmod 2;\\ 2, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 0\pmod 2;\end{array}\right. \end{eqnarray*} $\alpha \beta$ and $(\alpha +\beta)^2$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. The sequences $(u_n)_{n=0}^\infty$ are known as Lehmer sequences, and the terms of these sequences are known as Lehmer numbers. Lehmer showed that his sequences had similar divisibility properties to those of Lucas sequences, and he used them to extend the Lucas test for primality. \noindent We define a prime divisor $p$ of $u_n$ to be a primitive divisor of $u_n$ if $p$ does not divide $$(\alpha^2-\beta^2)^2u_3\cdots u_{n-1}.$$ Note that in the list of prime factors of the first $n-1$ terms of the sequence $(u_n)_{n=0}^\infty$, a primitive divisor of $u_n$ is a new prime factor. \noindent We let \begin{eqnarray*} \kappa& = & k(\alpha \beta\max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}),\\ \eta & = & \left\{\begin{array}{ll}1\hspace{.1in}\mbox{if}\hspace{.1in}\kappa\equiv 1\pmod 4,\\ 2\hspace{.1in}\mbox{otherwise},\end{array}\right. \end{eqnarray*} where $k(\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\})$ is the squarefree kernel of $\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}$. On the one hand, building on the work of Schinzel \cite{schinzelI}, we prove that if $n>4$, $n\neq 6$, $n/(\eta \kappa)$ is an odd integer, and the triple $(n,\alpha,\beta)$, in case $(\alpha-\beta)^2>0$, is not equivalent to a triple $(n,\alpha,\beta)$ from an explicit table, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. Moreover, we prove that if $n\geq 1.2\times 10^{10}$, and $n/(\eta \kappa)$ is an odd integer, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. On the other hand, building on the work of Stewart \cite{stewart77}, we prove that there are only finitely many triples $(n,\alpha,\beta)$, where $n>6$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, such that the $n$th Lehmer number $u_n$ has less than two primitive divisors, and that these triples may be explicitly determined. We determine all of these triples $(n,\alpha,\beta)$ up to equivalence explicitly when $6<n\leq 30$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, and we tabulate the triples $(n,\alpha,\beta)$ we discovered, up to equivalence, for $30<n\leq 500$. Finally, we show that the conditions $n>6$, $n\neq 12$, are best possible, subject to the truth of two plausible conjectures.

In 1878, Lucas \cite{lucas} investigated the sequences $(\ell_n)_{n=0}^\infty$ where $$\ell_n=\frac{\alpha^n-\beta^n}{\alpha-\beta},$$ $\alpha \beta$ and $\alpha+\beta$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. Lucas sequences are divisibility sequences; if $m|n$, then $\ell_m|\ell_n$, and more generally, $\gcd(\ell_m,\ell_n)=\ell_{\gcd(m,n)}$ for all positive integers $m$ and $n$. Matijasevic utilised this divisibility property of Lucas sequences in order to resolve Hilbert's 10th problem. \noindent In 1930, Lehmer \cite{lehmer} introduced the sequences $(u_n)_{n=0}^\infty$ where \begin{eqnarray*} u_n& = & \frac{\alpha^{n}-\beta^n}{\alpha^{\epsilon(n)}-\beta^{\epsilon(n)}},\\ \epsilon(n)&=&\left\{\begin{array}{ll} 1, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 1 \pmod 2;\\ 2, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 0\pmod 2;\end{array}\right. \end{eqnarray*} $\alpha \beta$ and $(\alpha +\beta)^2$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. The sequences $(u_n)_{n=0}^\infty$ are known as Lehmer sequences, and the terms of these sequences are known as Lehmer numbers. Lehmer showed that his sequences had similar divisibility properties to those of Lucas sequences, and he used them to extend the Lucas test for primality. \noindent We define a prime divisor $p$ of $u_n$ to be a primitive divisor of $u_n$ if $p$ does not divide $$(\alpha^2-\beta^2)^2u_3\cdots u_{n-1}.$$ Note that in the list of prime factors of the first $n-1$ terms of the sequence $(u_n)_{n=0}^\infty$, a primitive divisor of $u_n$ is a new prime factor. \noindent We let \begin{eqnarray*} \kappa& = & k(\alpha \beta\max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}),\\ \eta & = & \left\{\begin{array}{ll}1\hspace{.1in}\mbox{if}\hspace{.1in}\kappa\equiv 1\pmod 4,\\ 2\hspace{.1in}\mbox{otherwise},\end{array}\right. \end{eqnarray*} where $k(\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\})$ is the squarefree kernel of $\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}$. On the one hand, building on the work of Schinzel \cite{schinzelI}, we prove that if $n>4$, $n\neq 6$, $n/(\eta \kappa)$ is an odd integer, and the triple $(n,\alpha,\beta)$, in case $(\alpha-\beta)^2>0$, is not equivalent to a triple $(n,\alpha,\beta)$ from an explicit table, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. Moreover, we prove that if $n\geq 1.2\times 10^{10}$, and $n/(\eta \kappa)$ is an odd integer, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. On the other hand, building on the work of Stewart \cite{stewart77}, we prove that there are only finitely many triples $(n,\alpha,\beta)$, where $n>6$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, such that the $n$th Lehmer number $u_n$ has less than two primitive divisors, and that these triples may be explicitly determined. We determine all of these triples $(n,\alpha,\beta)$ up to equivalence explicitly when $6<n\leq 30$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, and we tabulate the triples $(n,\alpha,\beta)$ we discovered, up to equivalence, for $30<n\leq 500$. Finally, we show that the conditions $n>6$, $n\neq 12$, are best possible, subject to the truth of two plausible conjectures.