Pub Date : 2022-12-05DOI: 10.7546/nntdm.2022.28.4.794-795
A. Shannon
{"title":"Book review: “The Possibly True Story of Martin Gardiner”","authors":"A. Shannon","doi":"10.7546/nntdm.2022.28.4.794-795","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.794-795","url":null,"abstract":"","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42435775","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 : 2022-11-30DOI: 10.7546/nntdm.2022.28.4.765-777
B. Kuloğlu, E. Özkan
We introduce new kinds of k-Pell and k-Pell–Lucas numbers related to the distance between numbers by a recurrence relation and show their relation to the (k,r)-Pell and (k,r)-Pell–Lucas numbers. These sequences differ both according to the value of the natural number k and the value of a new parameter r in the definition of this distance. We give several properties of these sequences. In addition, we establish the generating functions, some important identities, as well as the sum of the terms of the generalized (k,r)-Pell and (k,r)-Pell–Lucas numbers. Furthermore, we indicate another way to obtain the generalized (k,r)-Pell and (k,r)-Pell–Lucas sequences from the generating function, in connection to graphs.
{"title":"On generalized (k, r)-Pell and (k, r)-Pell–Lucas numbers","authors":"B. Kuloğlu, E. Özkan","doi":"10.7546/nntdm.2022.28.4.765-777","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.765-777","url":null,"abstract":"We introduce new kinds of k-Pell and k-Pell–Lucas numbers related to the distance between numbers by a recurrence relation and show their relation to the (k,r)-Pell and (k,r)-Pell–Lucas numbers. These sequences differ both according to the value of the natural number k and the value of a new parameter r in the definition of this distance. We give several properties of these sequences. In addition, we establish the generating functions, some important identities, as well as the sum of the terms of the generalized (k,r)-Pell and (k,r)-Pell–Lucas numbers. Furthermore, we indicate another way to obtain the generalized (k,r)-Pell and (k,r)-Pell–Lucas sequences from the generating function, in connection to graphs.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44587596","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 : 2022-11-16DOI: 10.7546/nntdm.2022.28.4.758-764
K. Atanassov, Lilija Atanassova, A. Shannon
The term ‘combined’ sequence includes any of the ‘coupled’, ‘intercalated’ and ‘pulsated’ sequences. In this paper, k = 3, so new combined 3-Fibonacci sequences, {alpha_n }, { beta_n }, { gamma_n }, are introduced and the explicit formulae for their general terms are developed. That is, there are three such sequences, each with a linear recurrence relation which contains terms from the other two. In effect, each such recurrence relation is second order, with two initial terms which specify the subsequent delineation of the terms of the sequences. The initial terms are, respectively, langle alpha_0, alpha_1 rangle = langle 2a, 2d rangle, langle beta_0, beta_1 rangle = langle b,e rangle and langle gamma_0, gamma_1 rangle = langle 2c, 2f rangle in turn. These result in neat inter-relationships among the three sequences, which can lead to intriguing connections with known sequences, and to a surprisingly simple graphical representation of the whole process. The references include a comprehensive cover of the pertinent literature on these aspects of recursive sequences particularly during the last seventy years. A secondary goal of the paper is to put the disarray of this part of number theory into some semblance of order with a selection of representative references. This gives rise to a ‘combobulated sequence’, so-called because it restores partial order to a disarray of many papers into three classes, which are fuzzy in both their membership and non-membership because of their diverse and non-systematic derivations.
{"title":"On combined 3-Fibonacci sequences","authors":"K. Atanassov, Lilija Atanassova, A. Shannon","doi":"10.7546/nntdm.2022.28.4.758-764","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.758-764","url":null,"abstract":"The term ‘combined’ sequence includes any of the ‘coupled’, ‘intercalated’ and ‘pulsated’ sequences. In this paper, k = 3, so new combined 3-Fibonacci sequences, {alpha_n }, { beta_n }, { gamma_n }, are introduced and the explicit formulae for their general terms are developed. That is, there are three such sequences, each with a linear recurrence relation which contains terms from the other two. In effect, each such recurrence relation is second order, with two initial terms which specify the subsequent delineation of the terms of the sequences. The initial terms are, respectively, langle alpha_0, alpha_1 rangle = langle 2a, 2d rangle, langle beta_0, beta_1 rangle = langle b,e rangle and langle gamma_0, gamma_1 rangle = langle 2c, 2f rangle in turn. These result in neat inter-relationships among the three sequences, which can lead to intriguing connections with known sequences, and to a surprisingly simple graphical representation of the whole process. The references include a comprehensive cover of the pertinent literature on these aspects of recursive sequences particularly during the last seventy years. A secondary goal of the paper is to put the disarray of this part of number theory into some semblance of order with a selection of representative references. This gives rise to a ‘combobulated sequence’, so-called because it restores partial order to a disarray of many papers into three classes, which are fuzzy in both their membership and non-membership because of their diverse and non-systematic derivations.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45962635","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 : 2022-11-11DOI: 10.7546/nntdm.2022.28.4.744-748
Brahim Mittou
Let $pgeq3$ be a prime number and let $m, n$ and $l$ be integers with $gcd(l,p)=1$. Let $chi$ be a Dirichlet character modulo $p$ and $L(s,chi)$ be the Dirichlet L-function corresponding to $chi$. Explicit formulas for: $$dfrac{2}{p-1} sum limitssb{underset{chi(-1)=+1}{chihspace{-0.2cm} mod p}} chi(l) L(m,chi)L(n,overline{chi}) text{ and }dfrac{2}{p-1} sum limitssb{underset{chi(-1)=-1}{chihspace{-0.2cm} mod p}} chi(l) L(m,chi)L(n,overline{chi})$$ are given in this paper by using the properties of character sums and Bernoulli polynomials.
设$pgeq3$为质数，设$m, n$和$l$为整数，设$gcd(l,p)=1$为整数。设$chi$为狄利克雷字符模$p$, $L(s,chi)$为对应$chi$的狄利克雷l函数。本文利用特征和和伯努利多项式的性质，给出了:$$dfrac{2}{p-1} sum limitssb{underset{chi(-1)=+1}{chihspace{-0.2cm} mod p}} chi(l) L(m,chi)L(n,overline{chi}) text{ and }dfrac{2}{p-1} sum limitssb{underset{chi(-1)=-1}{chihspace{-0.2cm} mod p}} chi(l) L(m,chi)L(n,overline{chi})$$的显式表达式。
{"title":"Explicit formulas for sums related to Dirichlet L-functions","authors":"Brahim Mittou","doi":"10.7546/nntdm.2022.28.4.744-748","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.744-748","url":null,"abstract":"Let $pgeq3$ be a prime number and let $m, n$ and $l$ be integers with $gcd(l,p)=1$. Let $chi$ be a Dirichlet character modulo $p$ and $L(s,chi)$ be the Dirichlet L-function corresponding to $chi$. Explicit formulas for: $$dfrac{2}{p-1} sum limitssb{underset{chi(-1)=+1}{chihspace{-0.2cm} mod p}} chi(l) L(m,chi)L(n,overline{chi}) text{ and }dfrac{2}{p-1} sum limitssb{underset{chi(-1)=-1}{chihspace{-0.2cm} mod p}} chi(l) L(m,chi)L(n,overline{chi})$$ are given in this paper by using the properties of character sums and Bernoulli polynomials.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47288848","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 : 2022-11-07DOI: 10.7546/nntdm.2022.28.4.730-743
Samer Seraj
Suppose every integer is taken to the power of a fixed integer exponent k ≥ 2 and the remainders of these powers upon division by a fixed integer n ≥ 2 are found. It is natural to ask how many distinct remainders are produced. By building on the work of Stangl, who published the k = 2 case in Mathematics Magazine in 1996, we find essentially closed formulas that allow for the computation of this number for any k. Along the way, we provide an exposition of classical results on the multiplicativity of this counting function and results on the number of remainders that are coprime to the modulus n.
{"title":"Counting general power residues","authors":"Samer Seraj","doi":"10.7546/nntdm.2022.28.4.730-743","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.730-743","url":null,"abstract":"Suppose every integer is taken to the power of a fixed integer exponent k ≥ 2 and the remainders of these powers upon division by a fixed integer n ≥ 2 are found. It is natural to ask how many distinct remainders are produced. By building on the work of Stangl, who published the k = 2 case in Mathematics Magazine in 1996, we find essentially closed formulas that allow for the computation of this number for any k. Along the way, we provide an exposition of classical results on the multiplicativity of this counting function and results on the number of remainders that are coprime to the modulus n.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42486496","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}
{"title":"On Vandiver’s arithmetical function – II","authors":"J. Sándor","doi":"10.7546/nntdm.2022.28.4.710-718","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.710-718","url":null,"abstract":"We study more properties of Vandiver’s arithmetical function [V(n) = displaystyle prod_{dmid n} (d+1),] introduced in [2].","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44421372","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 : 2022-11-07DOI: 10.7546/nntdm.2022.28.4.719-729
A. Benyattou
In this paper, we use the properties of the classical umbral calculus to give some congruences related to the Bell numbers and Bell polynomials. We also present a new congruence involving Appell polynomials with integer coefficients.
{"title":"Congruences via umbral calculus","authors":"A. Benyattou","doi":"10.7546/nntdm.2022.28.4.719-729","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.719-729","url":null,"abstract":"In this paper, we use the properties of the classical umbral calculus to give some congruences related to the Bell numbers and Bell polynomials. We also present a new congruence involving Appell polynomials with integer coefficients.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49121107","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 : 2022-10-31DOI: 10.7546/nntdm.2022.28.4.698-709
R. Ndubuisi, U. K. Nwajeri, C. P. Onyenegecha, K. Patil, O. G. Udoaka, W. Osuji
This paper considers linear mappings in paraletrix spaces as an extension of the one given for rhotrix vector spaces. Furthermore, the adjoints of these mappings are given with their application in fractional calculus.
{"title":"Linear mappings in paraletrix spaces and their application to fractional calculus","authors":"R. Ndubuisi, U. K. Nwajeri, C. P. Onyenegecha, K. Patil, O. G. Udoaka, W. Osuji","doi":"10.7546/nntdm.2022.28.4.698-709","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.698-709","url":null,"abstract":"This paper considers linear mappings in paraletrix spaces as an extension of the one given for rhotrix vector spaces. Furthermore, the adjoints of these mappings are given with their application in fractional calculus.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44013432","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 : 2022-10-27DOI: 10.7546/nntdm.2022.28.4.692-697
V. Siva Rama Prasad, P. Anantha Reddy
Using a strictly increasing function $alpha: left [ 1,infty right )rightarrow left [ 1,infty right ),$ we define below (see(1.1) and (1.2)) two functions $S_{alpha}:left [ 1,infty right )rightarrow mathbb{N}$ and $S_{alpha}^*:left [ 1,infty right )rightarrow mathbb{N}$, where $mathbb{N}$ is the set of all natural numbers. The functions $S_{alpha}$ and $S_{alpha}^*$ respectively generalize the functions $S$ and $S_{*}$ introduced and studied by J. Sándor [5] as well as the functions $G$ and $G_{*}$ considered by N. Anitha [1]. In this paper we obtain several properties of $S_{alpha}$ and $S_{alpha}^*$ - some of which give the results of Sándor [5] and of Anitha [1] as special cases.
使用严格递增函数$alpha: left [ 1,infty right )rightarrow left [ 1,infty right ),$，我们在下面定义(见(1.1)和(1.2))两个函数$S_{alpha}:left [ 1,infty right )rightarrow mathbb{N}$和$S_{alpha}^*:left [ 1,infty right )rightarrow mathbb{N}$，其中$mathbb{N}$是所有自然数的集合。函数$S_{alpha}$和$S_{alpha}^*$分别推广了J. Sándor[5]介绍和研究的函数$S$和$S_{*}$，以及N. anita[1]考虑的函数$G$和$G_{*}$。本文得到了$S_{alpha}$和$S_{alpha}^*$的几个性质，其中一些性质给出了Sándor[5]和Anitha[1]作为特例的结果。
{"title":"On a generalization of a function of J. Sándor","authors":"V. Siva Rama Prasad, P. Anantha Reddy","doi":"10.7546/nntdm.2022.28.4.692-697","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.692-697","url":null,"abstract":"Using a strictly increasing function $alpha: left [ 1,infty right )rightarrow left [ 1,infty right ),$ we define below (see(1.1) and (1.2)) two functions $S_{alpha}:left [ 1,infty right )rightarrow mathbb{N}$ and $S_{alpha}^*:left [ 1,infty right )rightarrow mathbb{N}$, where $mathbb{N}$ is the set of all natural numbers. The functions $S_{alpha}$ and $S_{alpha}^*$ respectively generalize the functions $S$ and $S_{*}$ introduced and studied by J. Sándor [5] as well as the functions $G$ and $G_{*}$ considered by N. Anitha [1]. In this paper we obtain several properties of $S_{alpha}$ and $S_{alpha}^*$ - some of which give the results of Sándor [5] and of Anitha [1] as special cases.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45856694","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 : 2022-10-27DOI: 10.7546/nntdm.2022.28.4.677-691
P. Bush, K. V. Murphy
In this paper we investigate equations featuring sums of consecutive square integers, such as $3^2 + 4^2 = 5^2$, and $108^2 + 109^2 + 110^2 = 133^2 + 134^2$. In general, for a sum of $m+1$ consecutive square integers, $x^2 + (x+1)^2 + cdots + (x+m)^2$, there is a distinct set of $m$ consecutive squares, $(x+n)^2 + (x+(n+1))^2 + cdots + (x+(n+(m-1)))^2$, to which these are equal. We present a bootstrap method for constructing these equations, which yields solutions comprising an infinite two-dimensional array. We apply a similar method to constructing consecutive square sum equations involving $m+2$ terms on the left, and $m$ terms on the right, formed from two distinct sets of consecutive squares separated one term to the left of the equals sign, such as $2^2 + 3^2 + 6^2 = 7^2$.
{"title":"Equations of two sets of consecutive square sums","authors":"P. Bush, K. V. Murphy","doi":"10.7546/nntdm.2022.28.4.677-691","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.677-691","url":null,"abstract":"In this paper we investigate equations featuring sums of consecutive square integers, such as $3^2 + 4^2 = 5^2$, and $108^2 + 109^2 + 110^2 = 133^2 + 134^2$. In general, for a sum of $m+1$ consecutive square integers, $x^2 + (x+1)^2 + cdots + (x+m)^2$, there is a distinct set of $m$ consecutive squares, $(x+n)^2 + (x+(n+1))^2 + cdots + (x+(n+(m-1)))^2$, to which these are equal. We present a bootstrap method for constructing these equations, which yields solutions comprising an infinite two-dimensional array. We apply a similar method to constructing consecutive square sum equations involving $m+2$ terms on the left, and $m$ terms on the right, formed from two distinct sets of consecutive squares separated one term to the left of the equals sign, such as $2^2 + 3^2 + 6^2 = 7^2$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46733722","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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