Pub Date : 2023-12-29DOI: 10.7546/nntdm.2023.29.4.861-880
K. Atanassov
The concept of the object called “tertion” is discussed. Some operations over tertions are introduced and their properties are studied. The relationship between tertions, complex numbers are quaternions are discussed.
{"title":"On tertions and other algebraic objects","authors":"K. Atanassov","doi":"10.7546/nntdm.2023.29.4.861-880","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.861-880","url":null,"abstract":"The concept of the object called “tertion” is discussed. Some operations over tertions are introduced and their properties are studied. The relationship between tertions, complex numbers are quaternions are discussed.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"109 5","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139146970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.7546/nntdm.2023.29.4.813-819
K. Atanassov, József Sándor
A modification of the set $underline{rm Set}(n)$ for a fixed natural number $n$ is introduced in the form: $underline{rm Set}(n, f)$, where $f$ is an arithmetic function. The sets $underline{rm Set}(n,varphi), underline{rm Set}(n,psi), underline{rm Set}(n,sigma)$ are discussed, where $varphi, psi$ and $sigma$ are Euler's function, Dedekind's function and the sum of the positive divisors of $n$, respectively.
{"title":"On a modification of $underline{Set}(n)$","authors":"K. Atanassov, József Sándor","doi":"10.7546/nntdm.2023.29.4.813-819","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.813-819","url":null,"abstract":"A modification of the set $underline{rm Set}(n)$ for a fixed natural number $n$ is introduced in the form: $underline{rm Set}(n, f)$, where $f$ is an arithmetic function. The sets $underline{rm Set}(n,varphi), underline{rm Set}(n,psi), underline{rm Set}(n,sigma)$ are discussed, where $varphi, psi$ and $sigma$ are Euler's function, Dedekind's function and the sum of the positive divisors of $n$, respectively.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"220 4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138621081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.7546/nntdm.2023.29.4.827-841
E. Mehraban, Ö. Deveci, E. Hıncal
In this paper, we consider the 2-generator p-groups of nilpotency class 2. We will discuss the lengths of the periods of the t-Fibonacci sequences in these groups.
在本文中，我们考虑的是零能级 2 的 2 个生成器 p 群。我们将讨论这些群中 t-Fibonacci 序列的周期长度。
{"title":"The t-Fibonacci sequences in the 2-generator p-groups of nilpotency class 2","authors":"E. Mehraban, Ö. Deveci, E. Hıncal","doi":"10.7546/nntdm.2023.29.4.827-841","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.827-841","url":null,"abstract":"In this paper, we consider the 2-generator p-groups of nilpotency class 2. We will discuss the lengths of the periods of the t-Fibonacci sequences in these groups.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"125 6","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139191898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-30DOI: 10.7546/nntdm.2023.29.4.794-803
Volkan Aşkin
The undirected $P(Z_n)$ power graph of a finite group of $Z_n$ is a connected graph, the set of vertices of which is $Z_n$. Here $langle u, vrangle in P(Z_n)$ are two diverse adjacent vertices if and only if $u ne v$ and $langle v rangle subseteq langle u rangle$ or $langle u rangle subseteq langle v rangle$. We will shortly name the undirected $P(Z_n)$ power graph as the power graph $P(Z_n)$. The Wiener, hyper-Wiener, Harary and SK indices of the $P(Z_n)$ power graph are in order as follows $$frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },dleft( u,v right), frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },dleft( u,v right)+frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },{{d}^{2}}left(u,v right),$$ $$underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },frac{1}{dleft(u,v right)} mbox{ and } frac{1}{2}underset{uvin Eleft( G right)}{mathop sum },left( {{d}_{u}}+{{d}_{v}} right).$$ In this article we focus more on the indices of $P(Z_n)$ power graph by Wiener, hyper-Wiener, Harary and SK the definition of the power graph is presented and the results and theorems which we need in our discussion are provided in the introduction. Finally, the main point of the article is that we calculate the Wiener, hyper-Wiener, Harary and SK indices of the power graph $P(Z_n)$ corresponding to the vertex $n = p^k cdot q^r$. These are as follows: $p, q$ are distinct primes and $k, r$ are nonnegative integers.
$Z_n$的有限群的无向$P(Z_n)$幂图是一个连通图，其顶点集合为$Z_n$。这里，P(Z_n)$中的 $ulangle u, vrangle 是两个不同的相邻顶点，当且仅当 $u ne v$ 和 $ulangle v rangle subseteq langle u rangle$ 或 $ulangle u rangle subseteq langle v rangle$。我们很快会把不定向的 $P(Z_n)$ 幂图命名为幂图 $P(Z_n)$。$P(Z_n)$ 幂图的维纳指数、超维纳指数、哈拉里指数和 SK 指数依次为 $$frac{1}{2}underset{left{ u、v right}subseteq Vleft( G right)}{mathop sum },dleft( u,v right),frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum }、dleft( u,v right)+frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },{{d}^{2}}left(u. v right), $$$ Vleft( G right)}{mathop sum }、v right),$$$underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },frac{1}{dleft(u,v right)} mbox{ and }frac{1}{2}underset{uvin Eleft( Gright)}{mathop sum }left( {{d}_{u}}+{{d}_{v}} right).$$ 在本文中，我们将更多地关注维纳、超维纳、哈拉里和 SK 的 $P(Z_n)$ 幂图的指数，并在引言中介绍了幂图的定义以及讨论中所需的结果和定理。最后，文章的重点是我们计算与顶点 $n = p^k cdot q^r$ 相对应的幂图 $P(Z_n)$的维纳、超维纳、哈拉里和 SK 指数。这些指数如下：$p, q$ 为不同的素数，$k, r$ 为非负整数。
{"title":"The Wiener, hyper-Wiener, Harary and SK indices of the P(Z_{p^k·q^r}) power graph","authors":"Volkan Aşkin","doi":"10.7546/nntdm.2023.29.4.794-803","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.794-803","url":null,"abstract":"The undirected $P(Z_n)$ power graph of a finite group of $Z_n$ is a connected graph, the set of vertices of which is $Z_n$. Here $langle u, vrangle in P(Z_n)$ are two diverse adjacent vertices if and only if $u ne v$ and $langle v rangle subseteq langle u rangle$ or $langle u rangle subseteq langle v rangle$. We will shortly name the undirected $P(Z_n)$ power graph as the power graph $P(Z_n)$. The Wiener, hyper-Wiener, Harary and SK indices of the $P(Z_n)$ power graph are in order as follows $$frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },dleft( u,v right), frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },dleft( u,v right)+frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },{{d}^{2}}left(u,v right),$$ $$underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },frac{1}{dleft(u,v right)} mbox{ and } frac{1}{2}underset{uvin Eleft( G right)}{mathop sum },left( {{d}_{u}}+{{d}_{v}} right).$$ In this article we focus more on the indices of $P(Z_n)$ power graph by Wiener, hyper-Wiener, Harary and SK the definition of the power graph is presented and the results and theorems which we need in our discussion are provided in the introduction. Finally, the main point of the article is that we calculate the Wiener, hyper-Wiener, Harary and SK indices of the power graph $P(Z_n)$ corresponding to the vertex $n = p^k cdot q^r$. These are as follows: $p, q$ are distinct primes and $k, r$ are nonnegative integers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"67 5","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139205678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-30DOI: 10.7546/nntdm.2023.29.4.804-812
S. Koparal, N. Ömür, Laid Elkhiri
In this paper, we define generalized hyperharmonic numbers of order $r, H_{n,m}^{r}left( sigma right) ,$ for $min mathbb{Z}^{+}$ and give some applications by using generating functions of these numbers. For example, for $n, r, sin mathbb{Z}^{+}$ such that $1leq sleq r,$ begin{equation*} sumlimits_{k=1}^{n}binom{n-k+s-1}{s-1}H_{k,m}^{r-s}left( sigma right) =H_{n,m}^{r}left( sigma right), end{equation*} and begin{equation*} sum_{k=1}^{n}sum_{i=1}^{k}frac{H_{k-i,m}^{r+1}left( sigma right) D_{r}(k-i+r)}{(n-k)!left( k-i+rright) !}=H_{n,m}^{2r+2}(sigma ), end{equation*} where $D_{r}(n)$ is an $r$-derangement number.
{"title":"On generalized hyperharmonic numbers of order r, H_{n,m}^{r} (sigma)","authors":"S. Koparal, N. Ömür, Laid Elkhiri","doi":"10.7546/nntdm.2023.29.4.804-812","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.804-812","url":null,"abstract":"In this paper, we define generalized hyperharmonic numbers of order $r, H_{n,m}^{r}left( sigma right) ,$ for $min mathbb{Z}^{+}$ and give some applications by using generating functions of these numbers. For example, for $n, r, sin mathbb{Z}^{+}$ such that $1leq sleq r,$ begin{equation*} sumlimits_{k=1}^{n}binom{n-k+s-1}{s-1}H_{k,m}^{r-s}left( sigma right) =H_{n,m}^{r}left( sigma right), end{equation*} and begin{equation*} sum_{k=1}^{n}sum_{i=1}^{k}frac{H_{k-i,m}^{r+1}left( sigma right) D_{r}(k-i+r)}{(n-k)!left( k-i+rright) !}=H_{n,m}^{2r+2}(sigma ), end{equation*} where $D_{r}(n)$ is an $r$-derangement number.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139200086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-30DOI: 10.7546/nntdm.2023.29.4.789-793
Lilija Atanassova, Velin Andonov
A new Fibonacci-type sequence from pulsated type is introduced. The explicit form of its members is given.
介绍了一种新的脉动型斐波那契数列。给出了其成员的明确形式。
{"title":"New Fibonacci-type pulsated sequences","authors":"Lilija Atanassova, Velin Andonov","doi":"10.7546/nntdm.2023.29.4.789-793","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.789-793","url":null,"abstract":"A new Fibonacci-type sequence from pulsated type is introduced. The explicit form of its members is given.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"120 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139200594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-27DOI: 10.7546/nntdm.2023.29.4.752-773
Anthony G. Shannon, P. Shiue, Shen C. Huang
This paper both extends and generalizes recently published properties which have been developed by many authors for elements of the Leonardo sequence in the context of second-order recursive sequences. It does this by considering the difference equation properties of the homogeneous Fibonacci sequence and the non-homogeneous properties of their Leonardo sequence counterparts. This produces a number of new identities associated with a generalized Leonardo sequence and its associated algorithm, as well as some combinatorial results which lead into elegant properties of hyper-Fibonacci numbers in contrast to their ordinary Fibonacci number analogues, and as a convolution of Fibonacci and Leonardo numbers.
{"title":"Notes on generalized and extended Leonardo numbers","authors":"Anthony G. Shannon, P. Shiue, Shen C. Huang","doi":"10.7546/nntdm.2023.29.4.752-773","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.752-773","url":null,"abstract":"This paper both extends and generalizes recently published properties which have been developed by many authors for elements of the Leonardo sequence in the context of second-order recursive sequences. It does this by considering the difference equation properties of the homogeneous Fibonacci sequence and the non-homogeneous properties of their Leonardo sequence counterparts. This produces a number of new identities associated with a generalized Leonardo sequence and its associated algorithm, as well as some combinatorial results which lead into elegant properties of hyper-Fibonacci numbers in contrast to their ordinary Fibonacci number analogues, and as a convolution of Fibonacci and Leonardo numbers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"98 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139230156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-27DOI: 10.7546/nntdm.2023.29.4.774-788
Eduard C. Taganap, Rainier D. Almuete
We show the existence of solutions to the n-rooks problem and n-queens problem on chessboards with hexagonal cells, problems equivalent to certain three and six direction riders on ordinary chessboards. Translating the problems into graph theory problems, we determine the independence number (maximum size of independent set) of rooks graph and queens graph. We consider the $n times n$ planar diamond-shaped H_n with hexagonal cells, and the board $H_n$ as a flat torus $T_n$. Here, a rook can execute moves on lines perpendicular to the six sides of the cell it is placed, and a queen can execute moves on those lines together with lines through the six corners of the cell it is placed.
我们证明了在有六边形单元的棋盘上 n 车问题和 n 皇后问题的解的存在性，这些问题相当于普通棋盘上某些三向和六向车的问题。将这些问题转化为图论问题，我们确定了车图和后图的独立数（独立集的最大大小）。我们考虑具有六边形单元的 $n times n$ 平面菱形 H_n，并将棋盘 $H_n$ 视为平面环形 $T_n$。在这里，车可以在垂直于它所放置的单元的六条边的直线上执行棋步，而后可以在这些直线和通过它所放置的单元的六个角的直线上执行棋步。
{"title":"n-Rooks and n-queens problem on planar and modular chessboards with hexagonal cells","authors":"Eduard C. Taganap, Rainier D. Almuete","doi":"10.7546/nntdm.2023.29.4.774-788","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.774-788","url":null,"abstract":"We show the existence of solutions to the n-rooks problem and n-queens problem on chessboards with hexagonal cells, problems equivalent to certain three and six direction riders on ordinary chessboards. Translating the problems into graph theory problems, we determine the independence number (maximum size of independent set) of rooks graph and queens graph. We consider the $n times n$ planar diamond-shaped H_n with hexagonal cells, and the board $H_n$ as a flat torus $T_n$. Here, a rook can execute moves on lines perpendicular to the six sides of the cell it is placed, and a queen can execute moves on those lines together with lines through the six corners of the cell it is placed.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139234926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-26DOI: 10.7546/nntdm.2023.29.4.737-751
Shubham
We define the $p$-adic valuation tree of a polynomial $f(x,y)$ $in$ $mathbb{Z}[x,y]$ by which we can find its $p$-adic valuation at any point. This work includes diverse $2$-adic valuation trees of certain degree-two polynomials in two variables. Among these, the $2$-adic valuation tree of $x^2+y^2$ is most interesting. We use the observations from these trees to study the $2$-adic valuation tree of the general degree-two polynomial in $2$ variables. We also study the $2$-adic valuation tree of the polynomial $x^2y+5$.
{"title":"The 2-adic valuation of the general degree-2 polynomial in 2 variables","authors":"Shubham","doi":"10.7546/nntdm.2023.29.4.737-751","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.737-751","url":null,"abstract":"We define the $p$-adic valuation tree of a polynomial $f(x,y)$ $in$ $mathbb{Z}[x,y]$ by which we can find its $p$-adic valuation at any point. This work includes diverse $2$-adic valuation trees of certain degree-two polynomials in two variables. Among these, the $2$-adic valuation tree of $x^2+y^2$ is most interesting. We use the observations from these trees to study the $2$-adic valuation tree of the general degree-two polynomial in $2$ variables. We also study the $2$-adic valuation tree of the polynomial $x^2y+5$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139235549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-21DOI: 10.7546/nntdm.2023.29.4.713-716
Stoyan Dimitrov
The inequalities varphi^2(n)+psi^2(n)+sigma^2(n) geq 3n^2+2n+3 , varphi(n)psi(n)+varphi(n)sigma(n)+sigma(n)psi(n) geq 3n^2+2n-1 connecting varphi(n), psi(n) and sigma(n)-functions are formulated and proved.
{"title":"Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n)","authors":"Stoyan Dimitrov","doi":"10.7546/nntdm.2023.29.4.713-716","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.713-716","url":null,"abstract":"The inequalities varphi^2(n)+psi^2(n)+sigma^2(n) geq 3n^2+2n+3 , varphi(n)psi(n)+varphi(n)sigma(n)+sigma(n)psi(n) geq 3n^2+2n-1 connecting varphi(n), psi(n) and sigma(n)-functions are formulated and proved.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139251422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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