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Sums of divisors on arithmetic progressions 算术级数上的除数之和
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-15 DOI: 10.1007/s10998-023-00566-x
Prapanpong Pongsriiam
{"title":"Sums of divisors on arithmetic progressions","authors":"Prapanpong Pongsriiam","doi":"10.1007/s10998-023-00566-x","DOIUrl":"https://doi.org/10.1007/s10998-023-00566-x","url":null,"abstract":"<p>For each <span>(sin {mathbb {R}})</span> and <span>(nin {mathbb {N}})</span>, let <span>(sigma _s(n) = sum _{dmid n}d^s)</span>. In this article, we study the number of sign changes in the difference <span>(sigma _s(an+b)-sigma _s(cn+d))</span> where <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, <i>s</i> are fixed, the vectors (<i>a</i>, <i>b</i>) and (<i>c</i>, <i>d</i>) are linearly independent over <span>({mathbb {Q}})</span>, and <i>n</i> runs over all positive integers. We also give several examples and propose some problems.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a class of Lebesgue-Ramanujan-Nagell equations 关于一类 Lebesgue-Ramanujan-Nagell 方程
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-14 DOI: 10.1007/s10998-023-00564-z
Azizul Hoque
{"title":"On a class of Lebesgue-Ramanujan-Nagell equations","authors":"Azizul Hoque","doi":"10.1007/s10998-023-00564-z","DOIUrl":"https://doi.org/10.1007/s10998-023-00564-z","url":null,"abstract":"<p>We deeply investigate the Diophantine equation <span>(cx^2+d^{2m+1}=2y^n)</span> in integers <span>(x, yge 1, mge 0)</span> and <span>(nge 3)</span>, where <i>c</i> and <i>d</i> are coprime positive integers satisfying <span>(cdnot equiv 3 pmod 4)</span>. We first solve this equation for prime <i>n</i> under the condition <span>(gcd (n, h(-cd))=1)</span>, where <span>(h(-cd))</span> denotes the class number of the imaginary quadratic field <span>({mathbb {Q}}(sqrt{-cd}))</span>. We then completely solve this equation for both <i>c</i> and <i>d</i> primes under the assumption <span>(gcd (n, h(-cd))=1)</span>. We also completely solve this equation for <span>(c=1)</span> and <span>(dequiv 1 pmod 4)</span> under the condition <span>(gcd (n, h(-d))=1)</span>. For some fixed values of <i>c</i> and <i>d</i>, we derive some results concerning the solvability of this equation.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extensions of a Diophantine triple by adjoining smaller elements II 通过邻接较小元素扩展 Diophantine 三元组 II
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-14 DOI: 10.1007/s10998-023-00569-8
Mihai Cipu, Andrej Dujella, Yasutsugu Fujita
{"title":"Extensions of a Diophantine triple by adjoining smaller elements II","authors":"Mihai Cipu, Andrej Dujella, Yasutsugu Fujita","doi":"10.1007/s10998-023-00569-8","DOIUrl":"https://doi.org/10.1007/s10998-023-00569-8","url":null,"abstract":"<p>Let <span>({a_1,b,c})</span> and <span>({a_2,b,c})</span> be Diophantine triples with <span>(a_1&lt;b&lt;a_2&lt;c)</span> and <span>(a_2ne b+c-2sqrt{bc+1})</span>. Put <span>(d_2=a_2+b+c+2a_2bc-2r_2st)</span>, where <span>(r_2=sqrt{a_2b+1})</span>, <span>(s=sqrt{ac+1})</span> and <span>(t=sqrt{bc+1})</span>. In this paper, we prove that if <span>(c le 16mu ^2 b^3)</span>, where <span>(mu =min {a_1,d_2})</span>, then <span>({a_1,a_2,b,c})</span> is a Diophantine quadruple. Combining this result with one of our previous results implies that if <span>({a_i,b,c,d})</span> <span>((iin {1,2,3}))</span> are Diophantine quadruples with <span>(a_1&lt;a_2&lt;b&lt;a_3&lt;c&lt;d)</span>, then <span>(a_3=b+c-2sqrt{bc+1})</span>. It immediately follows that there does not exist a septuple <span>({a_1,a_2,a_3,a_4,b,c,d})</span> with <span>(a_1&lt;a_2&lt;b&lt;a_3&lt;a_4&lt;c&lt;d)</span> such that <span>({a_i,b,c,d})</span> <span>((i in {1,2,3,4}))</span> are Diophantine quadruples. Moreover, it is shown that there are only finitely many sextuples <span>({a_1,a_2,a_3,b,c,d})</span> with <span>(a_1&lt;b&lt;a_2&lt;a_3&lt;c&lt;d)</span> such that <span>({a_i,b,c,d})</span> <span>((i in {1,2,3}))</span> are Diophantine quadruples.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The ascending chain condition on principal right ideals for semigroup constructions 半群构造的主权利理想的升链条件
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-14 DOI: 10.1007/s10998-023-00570-1
Craig Miller
{"title":"The ascending chain condition on principal right ideals for semigroup constructions","authors":"Craig Miller","doi":"10.1007/s10998-023-00570-1","DOIUrl":"https://doi.org/10.1007/s10998-023-00570-1","url":null,"abstract":"<p>We call a semigroup <span>({mathcal {R}})</span><i>-noetherian</i> if it satisfies the ascending chain condition on principal right ideals, or, equivalently, the ascending chain condition on <span>({mathcal {R}})</span>-classes. We investigate the behaviour of the property of being <span>({mathcal {R}}text {-noetherian})</span> under the following standard semigroup-theoretic constructions: semidirect products, Schützenberger products, free products, Rees matrix semigroups, Brandt extensions, Bruck–Reilly extensions and semilattices of semigroups.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138627975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On some maximal and minimal sets 关于一些最大集和最小集
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-12 DOI: 10.1007/s10998-023-00559-w
Jin-Hui Fang, Xue-Qin Cao
{"title":"On some maximal and minimal sets","authors":"Jin-Hui Fang, Xue-Qin Cao","doi":"10.1007/s10998-023-00559-w","DOIUrl":"https://doi.org/10.1007/s10998-023-00559-w","url":null,"abstract":"<p>A set <i>A</i> of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such <i>A</i> is called <i>maximal</i> if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3-free set <span>({a_1&lt;a_2&lt;cdots&lt;a_n&lt;cdots })</span> of positive integers with the property that <span>(lim _{nrightarrow infty }(a_{n+1}-a_n)=infty )</span>. In this paper, we generalize their result. On the other hand, a set <i>A</i> of nonnegative integers is called an asymptotic basis of order <i>h</i> if every sufficiently large integer can be represented as a sum of <i>h</i> elements of <i>A</i>. Such <i>A</i> is defined as <i>minimal</i> if no proper subset of <i>A</i> has this property. We also extend a result of Jańczak and Schoen about the minimal asymptotic basis.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138580311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Greenberg’s conjecture for certain real biquadratic fields 关于格林伯格对某些实双曲域的猜想
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-09 DOI: 10.1007/s10998-023-00560-3
Abdelakder El Mahi, M’hammed Ziane
{"title":"On Greenberg’s conjecture for certain real biquadratic fields","authors":"Abdelakder El Mahi, M’hammed Ziane","doi":"10.1007/s10998-023-00560-3","DOIUrl":"https://doi.org/10.1007/s10998-023-00560-3","url":null,"abstract":"<p>In this paper, we give the structure of the Iwasawa module <span>(X=X(k_{infty }))</span> of the <span>(mathbb {Z}_{2})</span>-extension of infinitely many real biquadratic fields <i>k</i>. Denote by <span>(lambda , mu )</span> and <span>(nu )</span> the Iwasawa invariants of the cyclotomic <span>(mathbb {Z}_{2})</span>-extension of <i>k</i>. Then <span>(lambda =mu =0 )</span> and <span>(nu =2)</span>.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138560445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The c-differential uniformity of the perturbed inverse function via a trace function $$ {{,textrm{Tr},}}big (frac{x^2}{x+1}big )$$ 通过痕量函数计算扰动反函数的 c 微分均匀性 $$ {{,textrm{Tr},}}big (frac{x^2}{x+1}big )$$
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-09 DOI: 10.1007/s10998-023-00561-2
Kübra Kaytancı, Ferruh Özbudak
{"title":"The c-differential uniformity of the perturbed inverse function via a trace function $$ {{,textrm{Tr},}}big (frac{x^2}{x+1}big )$$","authors":"Kübra Kaytancı, Ferruh Özbudak","doi":"10.1007/s10998-023-00561-2","DOIUrl":"https://doi.org/10.1007/s10998-023-00561-2","url":null,"abstract":"<p>Differential uniformity is one of the most crucial concepts in cryptography. Recently Ellingsen et al. (IEEE Trans Inf Theory 66:5781–5789, 2020) introduced a new concept, the c-Difference Distribution Table and the c-differential uniformity, by extending the usual differential notion. The motivation behind this new concept is based on having the ability to resist some known differential attacks which is shown by Borisov et. al. (Multiplicative Differentials, 2002). In 2022, Hasan et al. (IEEE Trans Inf Theory 68:679–691, 2022) gave an upper bound of the c-differential uniformity of the perturbed inverse function <i>H</i> via a trace function <span>( {{,textrm{Tr},}}big (frac{x^2}{x+1}big ))</span>. In their work, they also presented an open question on the exact c-differential uniformity of <i>H</i>. By using a new method based on algebraic curves over finite fields, we solve the open question in the case <span>( {{,textrm{Tr},}}(c)=1= {{,textrm{Tr},}}(frac{1}{c}))</span> for <span>( c in {mathbb {F}}_{2^n}setminus {0,1} )</span> completely and we show that the exact c-differential uniformity of <i>H</i> is 8. In the remaining case, we almost completely solve the problem and we show that the c-differential uniformity of <i>H</i> is either 8 or 9.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138560260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Three Problems of Y.–G. Chen 论陈永刚的三个问题
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-07 DOI: 10.1007/s10998-023-00563-0
Yuchen Ding
{"title":"On Three Problems of Y.–G. Chen","authors":"Yuchen Ding","doi":"10.1007/s10998-023-00563-0","DOIUrl":"https://doi.org/10.1007/s10998-023-00563-0","url":null,"abstract":"<p>In this short note, we answer two questions of Chen and Ruzsa negatively and answer a question of Ma and Chen affirmatively.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A degenerate Kirchhoff-type problem involving variable $$s(cdot )$$ -order fractional $$p(cdot )$$ -Laplacian with weights 涉及带权重的变量$$s(cdot )$$ -阶分数$$p(cdot )$$ -拉普拉奇的退化基尔霍夫型问题
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-12-07 DOI: 10.1007/s10998-023-00562-1
Mostafa Allaoui, Mohamed Karim Hamdani, Lamine Mbarki
{"title":"A degenerate Kirchhoff-type problem involving variable $$s(cdot )$$ -order fractional $$p(cdot )$$ -Laplacian with weights","authors":"Mostafa Allaoui, Mohamed Karim Hamdani, Lamine Mbarki","doi":"10.1007/s10998-023-00562-1","DOIUrl":"https://doi.org/10.1007/s10998-023-00562-1","url":null,"abstract":"<p>This paper deals with a class of nonlocal variable <i>s</i>(.)-order fractional <i>p</i>(.)-Kirchhoff type equations: </p><span>$$begin{aligned} left{ begin{array}{ll} {mathcal {K}}left( int _{{mathbb {R}}^{2N}}frac{1}{p(x,y)}frac{|varphi (x)-varphi (y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}} ,dx,dyright) (-Delta )^{s(cdot )}_{p(cdot )}varphi (x) =f(x,varphi ) quad text{ in } Omega , varphi =0 quad text{ on } {mathbb {R}}^Nbackslash Omega . end{array} right. end{aligned}$$</span><p>Under some suitable conditions on the functions <span>(p,s, {mathcal {K}})</span> and <i>f</i>, the existence and multiplicity of nontrivial solutions for the above problem are obtained. Our results cover the degenerate case in the <span>(p(cdot ))</span> fractional setting.\u0000</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A regular Tóth identity and a Menon-type identity in residually finite Dedekind domains 剩余有限Dedekind域中的正则Tóth恒等式和menon型恒等式
IF 0.8 3区 数学
Periodica Mathematica Hungarica Pub Date : 2023-11-24 DOI: 10.1007/s10998-023-00555-0
Tianfang Qi
{"title":"A regular Tóth identity and a Menon-type identity in residually finite Dedekind domains","authors":"Tianfang Qi","doi":"10.1007/s10998-023-00555-0","DOIUrl":"https://doi.org/10.1007/s10998-023-00555-0","url":null,"abstract":"<p>In this paper, we define the <i>s</i>-dimensional regular generalized Euler function and give a variant of Tóth’s identity in residually finite Dedekind domains, which can be viewed as a multidimensional version of the results by Wang, Zhang, Ji (2019) and Ji, Wang (2020).</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","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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