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Congruences for harmonic sums 调和和的同余
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-03-01 DOI: 10.7546/nntdm.2023.29.1.137-146
Yining Yang, Peng Yang
{"title":"Congruences for harmonic sums","authors":"Yining Yang, Peng Yang","doi":"10.7546/nntdm.2023.29.1.137-146","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.137-146","url":null,"abstract":"Zhao found a curious congruence modulo p on harmonic sums. Xia and Cai generalized his congruence to a supercongruence modulo p^2. In this paper, we improve the harmonic sums [ H_{p}(n)=sumlimits_{substack{l_{1}+l_{2}+cdots+l_{n}=p l_{1}, l_{2}, ldots , l_{n}>0}} frac{1}{l_{1} l_{2} cdots l_{n}} ] to supercongruences modulo p^3 and p^4 for odd and even where prime p>8 and 3 leq n leq p-6.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45384772","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}
引用次数: 0
Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers 涉及二项式系数、第二类伯努利数和调和数的和
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-02-27 DOI: 10.7546/nntdm.2023.29.1.78-97
Necdet Batır, A. Sofo
{"title":"Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers","authors":"Necdet Batır, A. Sofo","doi":"10.7546/nntdm.2023.29.1.78-97","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.78-97","url":null,"abstract":"We offer a number of various finite and infinite sum identities involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. For example, among many others, we prove [displaystyle sum_{k=0}^{n}frac{(-1)^{k}h_{k}}{4^{k}} {{2k} choose {k}}G_{n-k}=frac{(-1)^{n-1}}{2^{2n-1}}{{2n-2} choose {n-1}}] and [displaystyle sum_{k=1}^{infty}frac{h_{k}}{k^{2}(2k-1)4^{k}} {{2k} choose {k}}=2pi +3zeta(2)log 2-3zeta(2)-frac{7}{2}zeta(3),] where h_k=H_{2k}-dfrac{1}{2}H_{k}, G_k are Bernoulli numbers of the second kind, and zeta is the Riemann zeta function. We also give an alternate proof of the series representations for the constants log (2 pi) and gamma given by Blagouchine and Coppo.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43296373","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}
引用次数: 0
Transcendental properties of the certain mix infinite products 某混合无穷乘积的超越性质
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-02-18 DOI: 10.7546/nntdm.2023.29.1.48-61
E. Miyanohara
{"title":"Transcendental properties of the certain mix infinite products","authors":"E. Miyanohara","doi":"10.7546/nntdm.2023.29.1.48-61","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.48-61","url":null,"abstract":"Let $k$ and $l$ be two multiplicatively independent positive integers and $b$ be an integer with $bge2$. Let $S$ be a finite set of integers. Nishioka proved that for any algebraic number $alpha$ with $0<|alpha|<1$ the infinite products $prod_{y=0}^{infty}(1-{alpha}^{d^{y}})$ ($d=2,3,ldots$) are algebraically independent over $mathbb{Q}$. As her result, for example, the transcendence of $prod_{y=0}^{infty}(1-frac{1}{{b}^{2^{y}}})prod_{y=0}^{infty}(1-frac{1}{{b}^{3^{y}}})$ is deduced. On the other hand, Tachiya, Amou–Väänänen investigated the certain infinite products which satisfy infinite chains of Mahler functional equation. The special case of the result of Tachiya shows that the infinite product $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})$ with $tau(i,y)in S$ ($1le ile k-1, yge0$) is either rational or transcendental. In this paper, we prove that the infinite product $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})prod_{yge0}(1+sum_{j=1}^{l-1} frac{delta(j,y)}{b^{jl^y}})$ with $tau(i,y),delta(j,y) in S$ $(1le ile k-1, 1le jle l-1, yge0)$ is either rational or transcendental. Moreover, we give sufficient conditions that $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})prod_{yge0}(1+sum_{j=1}^{l-1} frac{delta(j,y)}{b^{jl^y}})$ is transcendental.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48772527","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}
引用次数: 0
A study of the complexification process of the (s,t)-Perrin sequence (s,t)-Perrin序列络合过程的研究
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-02-14 DOI: 10.7546/nntdm.2023.29.1.40-47
R. Vieira, Francisco Regis Vieira Al, P. Catarino
{"title":"A study of the complexification process of the (s,t)-Perrin sequence","authors":"R. Vieira, Francisco Regis Vieira Al, P. Catarino","doi":"10.7546/nntdm.2023.29.1.40-47","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.40-47","url":null,"abstract":"The present article deals with the study of the generalized (s,t)-Perrin sequence in its complex process. Thus, from the one-dimensional model of the generalized (s,t)-Perrin sequence, imaginary units are inserted, starting with the insertion of unit i, called two-dimensional relations. Altogether, we have the n-dimensional relationships of the generalized (s,t)-Perrin sequence.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46234636","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}
引用次数: 0
Two generalizations of Liouville λ function Liouville λ函数的两种推广
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-02-13 DOI: 10.7546/nntdm.2023.29.1.30-39
A. P. Camargo
{"title":"Two generalizations of Liouville λ function","authors":"A. P. Camargo","doi":"10.7546/nntdm.2023.29.1.30-39","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.30-39","url":null,"abstract":"We study the properties of two classes of functions $lambda_k$ and $tilde{lambda}_k$ that generalize the Liouville $lambda$ function, including some equivalencies between the Riemann hypothesis and some assertions about the asymptotic behavior of the summatory functions of $lambda_k$ and $tilde{lambda}_k.$ Similar results are obtained for the generalization of the Möbius function considered by Tanaka.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44081319","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}
引用次数: 0
A note on the number $a^n+ b^n – dc^n$ 关于数字$ A ^n+ b^n - dc^n$的注释
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-02-08 DOI: 10.7546/nntdm.2023.29.1.24-29
N. Dung
{"title":"A note on the number $a^n+ b^n – dc^n$","authors":"N. Dung","doi":"10.7546/nntdm.2023.29.1.24-29","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.24-29","url":null,"abstract":"We say that a positive integer $d$ is special number of degree $n$ if for every integer $m$, there exist nonzero integers $a,b,c$ such that $m=a^n+b^n-dc^n$. In this paper, we investigate some necessary conditions on $n$ for existing a special number of degree $n$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45705478","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}
引用次数: 0
On distribution of the number of semisimple rings of order at most x in an arithmetic progression 等差数列中至多为x阶的半单环数的分布
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-02-06 DOI: 10.7546/nntdm.2023.29.1.17-23
Thorranin Thansri, T. Srichan, Pinthira Tangsupphathawat
{"title":"On distribution of the number of semisimple rings of order at most x in an arithmetic progression","authors":"Thorranin Thansri, T. Srichan, Pinthira Tangsupphathawat","doi":"10.7546/nntdm.2023.29.1.17-23","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.17-23","url":null,"abstract":"Let ell and q denote relatively prime positive integers. In this article, we derive the asymptotic formula for the summation begin{align*} sum_{nleq xatop nequiv ell pmod q}S(n), end{align*} where S(n) denotes the number of non-isomorphic finite semisimple rings with n elements.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48485426","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}
引用次数: 0
Pauli–Leonardo quaternions 保利-莱昂纳多四元数
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-01-25 DOI: 10.7546/nntdm.2023.29.1.1-16
Zehra İşbilir, M. Akyiğit, M. Tosun
{"title":"Pauli–Leonardo quaternions","authors":"Zehra İşbilir, M. Akyiğit, M. Tosun","doi":"10.7546/nntdm.2023.29.1.1-16","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.1-16","url":null,"abstract":"In this study, we define Pauli–Leonardo quaternions by taking the coefficients of the Pauli quaternions as Leonardo numbers. We give the recurrence relation, Binet formula, generating function, exponential generating function, some special equalities, and the sum properties of these novel quaternions. In addition, we investigate the interrelations between Pauli–Leonardo quaternions and the Pauli–Fibonacci, Pauli–Lucas quaternions. Moreover, we create some algorithms that determine the terms of the Pauli–Leonardo quaternions. Finally, we generate the matrix representations of the Pauli–Leonardo quaternions and ℝ-linear transformations.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46364295","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}
引用次数: 1
Combinatorial proofs of identities for the generalized Leonardo numbers 广义Leonardo数恒等式的组合证明
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2022-12-05 DOI: 10.7546/nntdm.2022.28.4.778-790
M. Shattuck
{"title":"Combinatorial proofs of identities for the generalized Leonardo numbers","authors":"M. Shattuck","doi":"10.7546/nntdm.2022.28.4.778-790","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.778-790","url":null,"abstract":"In this paper, we provide combinatorial proofs of several prior identities satisfied by the recently introduced generalized Leonardo numbers, denoted by mathcal{L}_{k,n}, as well as derive some new formulas. To do so, we interpret mathcal{L}_{k,n} as the enumerator of two classes of linear colored tilings of length n. A comparable treatment is also given for the incomplete generalized Leonardo numbers. Finally, a (p,q)-generalization of mathcal{L}_{k,n} is obtained by considering the joint distribution of a pair of statistics on one of the aforementioned classes of colored tilings.","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":"45528055","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}
引用次数: 3
In Memoriam: Prof. John Turner (1928 – 2022) 纪念:约翰·特纳教授(1928 - 2022)
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2022-12-05 DOI: 10.7546/nntdm.2022.28.4.791-793
A. Shannon
{"title":"In Memoriam: Prof. John Turner (1928 – 2022)","authors":"A. Shannon","doi":"10.7546/nntdm.2022.28.4.791-793","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.791-793","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":"45543092","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}
引用次数: 0
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