Pub Date : 2024-03-16DOI: 10.1142/s1793042124500477
Xing-Wang Jiang, Wu-Xia Ma
For a given sequence of nonnegative integers, let be the set of all finite subsequence sums of . is called complete if contains all sufficiently large integers. A real number is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of . Hegyvári conjectured that is complete if or is i.d.f. and , where is a sequence of integers. In this paper, we give a partial result of Hegyvári’s conjecture.
{"title":"A conjecture of Hegyvári","authors":"Xing-Wang Jiang, Wu-Xia Ma","doi":"10.1142/s1793042124500477","DOIUrl":"https://doi.org/10.1142/s1793042124500477","url":null,"abstract":"<p>For a given sequence <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span> of nonnegative integers, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>P</mi><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math></span><span></span> be the set of all finite subsequence sums of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span>. <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span> is called complete if <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>P</mi><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math></span><span></span> contains all sufficiently large integers. A real number <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi><mo>></mo><mn>0</mn></math></span><span></span> is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi></math></span><span></span>. Hegyvári conjectured that <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub></math></span><span></span> is complete if <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi></math></span><span></span> or <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>β</mi></math></span><span></span> is i.d.f. and <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi><mo stretchy=\"false\">/</mo><mi>β</mi><mo>≠</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>l</mi></mrow></msup><mspace width=\"0.25em\"></mspace><mo stretchy=\"false\">(</mo><mi>l</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, where <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub><mo>=</mo><mo stretchy=\"false\">{</mo><mo stretchy=\"false\">[</mo><mi>α</mi><mo stretchy=\"false\">]</mo><mo>,</mo><mo stretchy=\"false\">[</mo><mi>β</mi><mo stretchy=\"false\">]</mo><mo>,</mo><mo>…</mo><mo>,</mo><mo stretchy=\"false\">[</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>α</mi><mo stretchy=\"false\">]</mo><mo>,</mo><mo stretchy=\"false\">[</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>β</mi><mo stretchy=\"false\">]</mo><mo>,</mo><mo>…</mo><mo stretchy=\"false\">}</mo></math></span><span></span> is a sequence of integers. In this paper, we give a partial result of Hegyvári’s conjecture.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204335","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}
Pub Date : 2024-03-13DOI: 10.1142/s1793042124500519
Dominic Lanphier, Allen Lin
We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.
{"title":"Values of certain Dirichlet series and higher derivative formulas of trigonometric functions","authors":"Dominic Lanphier, Allen Lin","doi":"10.1142/s1793042124500519","DOIUrl":"https://doi.org/10.1142/s1793042124500519","url":null,"abstract":"<p>We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204257","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}
Pub Date : 2023-11-02DOI: 10.1142/s1793042124500179
Fang-Gang Xue
Let [Formula: see text] be the set of integers and [Formula: see text] the set of positive integers. For a nonempty set [Formula: see text] of integers and any integers [Formula: see text], [Formula: see text] with [Formula: see text], define [Formula: see text] as the number of solutions of [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] A set [Formula: see text] of integers is defined as a basis of order [Formula: see text] for [Formula: see text] if [Formula: see text] for every integer [Formula: see text]. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded bases. In this paper, we generalize the above result and obtain that: for any function [Formula: see text], there exists a bounded basis of order [Formula: see text] for [Formula: see text] such that [Formula: see text] for every integer [Formula: see text].
{"title":"On bounded basis with prescribed representation functions","authors":"Fang-Gang Xue","doi":"10.1142/s1793042124500179","DOIUrl":"https://doi.org/10.1142/s1793042124500179","url":null,"abstract":"Let [Formula: see text] be the set of integers and [Formula: see text] the set of positive integers. For a nonempty set [Formula: see text] of integers and any integers [Formula: see text], [Formula: see text] with [Formula: see text], define [Formula: see text] as the number of solutions of [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] A set [Formula: see text] of integers is defined as a basis of order [Formula: see text] for [Formula: see text] if [Formula: see text] for every integer [Formula: see text]. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded bases. In this paper, we generalize the above result and obtain that: for any function [Formula: see text], there exists a bounded basis of order [Formula: see text] for [Formula: see text] such that [Formula: see text] for every integer [Formula: see text].","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135875133","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}
Pub Date : 2023-11-02DOI: 10.1142/s1793042124500283
Archit Agarwal, Meghali Garg
Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler’s Diophantine equation [Formula: see text]. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation [Formula: see text] in the spirit of Ramanujan.
{"title":"Infinite families of solutions for A3 + B3 = C3 + D3 and A4 + B4 + C4 + D4 + E4 = F4 in the spirit of Ramanujan","authors":"Archit Agarwal, Meghali Garg","doi":"10.1142/s1793042124500283","DOIUrl":"https://doi.org/10.1142/s1793042124500283","url":null,"abstract":"Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler’s Diophantine equation [Formula: see text]. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation [Formula: see text] in the spirit of Ramanujan.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135875134","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}
Pub Date : 2023-11-02DOI: 10.1142/s1793042124500246
Zachary Feng
We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing [Formula: see text]. Applied to imaginary quadratic fields, this proportion is at least [Formula: see text]. Applied to cyclotomic fields [Formula: see text] with [Formula: see text], this proportion is at least [Formula: see text] with only finitely many exceptions of [Formula: see text], for any choice of [Formula: see text].
{"title":"A Lower Bound on the Proportion of Modular Elliptic Curves Over Galois CM Fields","authors":"Zachary Feng","doi":"10.1142/s1793042124500246","DOIUrl":"https://doi.org/10.1142/s1793042124500246","url":null,"abstract":"We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing [Formula: see text]. Applied to imaginary quadratic fields, this proportion is at least [Formula: see text]. Applied to cyclotomic fields [Formula: see text] with [Formula: see text], this proportion is at least [Formula: see text] with only finitely many exceptions of [Formula: see text], for any choice of [Formula: see text].","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135875328","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}
Pub Date : 2023-11-01DOI: 10.1142/s1793042123990014
{"title":"Author index (Volume 19)","authors":"","doi":"10.1142/s1793042123990014","DOIUrl":"https://doi.org/10.1142/s1793042123990014","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139302083","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500209
Alex Capunay
{"title":"Computing Shintani Domains","authors":"Alex Capunay","doi":"10.1142/s1793042124500209","DOIUrl":"https://doi.org/10.1142/s1793042124500209","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918481","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500180
Dazhao Tang
{"title":"Congruence properties modulo powers of 2 for overpartitions and overpartition pairs","authors":"Dazhao Tang","doi":"10.1142/s1793042124500180","DOIUrl":"https://doi.org/10.1142/s1793042124500180","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918640","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}
Pub Date : 2023-10-13DOI: 10.1142/s179304212450043x
Kui Liu, Jie Wu, Zhishan Yang
Denote by $tau$ k (n), $omega$(n) and $mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $omega$, 2 $omega$ , $mu$ 2 , $tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $epsilon$ (x $theta$ f +$epsilon$) for x $rightarrow$ $infty$, where $theta$ $omega$ = 53 110 , $theta$ 2 $omega$ = 9 19 , $theta$ $mu$2 = 2 5 , $theta$ $tau$ k = 5k--1 10k--1 and $epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{`e}s.
表示为 $tau$ K (n)， $omega$(n)及 $mu$ 2 (n) n作为k个自然数乘积的表示形式的个数，n的不同质因数的个数，以及无平方整数的特征函数。设[t]为实数t的积分部分，令f = $omega$， 2 $omega$ ， $mu$ 2、 $tau$ k，我们证明了n x f x n = x d1 f (d) d(d + 1) + 0 $epsilon$ (x) $theta$ F +$epsilon$) for x $rightarrow$ $infty$，其中 $theta$ $omega$ = 53 110， $theta$ 2 $omega$ = 9 19， $theta$ $mu$2 = 2 5， $theta$ $tau$ K = 5k- 1 10k- 1和 $epsilon$ > 0是一个任意小的正数。这些改进了Bordell的相应结果{è}5 .答案:
{"title":"On Some Sums Involving the Integral Part Function","authors":"Kui Liu, Jie Wu, Zhishan Yang","doi":"10.1142/s179304212450043x","DOIUrl":"https://doi.org/10.1142/s179304212450043x","url":null,"abstract":"Denote by $tau$ k (n), $omega$(n) and $mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $omega$, 2 $omega$ , $mu$ 2 , $tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $epsilon$ (x $theta$ f +$epsilon$) for x $rightarrow$ $infty$, where $theta$ $omega$ = 53 110 , $theta$ 2 $omega$ = 9 19 , $theta$ $mu$2 = 2 5 , $theta$ $tau$ k = 5k--1 10k--1 and $epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{`e}s.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135805960","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500362
Ertan Elma
{"title":"A Dirichlet Series Related to the Error Term in the Prime Number Theorem","authors":"Ertan Elma","doi":"10.1142/s1793042124500362","DOIUrl":"https://doi.org/10.1142/s1793042124500362","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918172","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}
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