International Journal of Number Theory最新文献

{"title":"Fourier coefficients of cusp forms on special sequences","authors":"Weili Yao","doi":"10.1142/s1793042124500568","DOIUrl":"https://doi.org/10.1142/s1793042124500568","url":null,"abstract":"<p>In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi></math></span><span></span> and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204341","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}

{"title":"Density questions in rings of the form 𝒪K[γ] ∩ K","authors":"Deepesh Singhal, Yuxin Lin","doi":"10.1142/s1793042124500581","DOIUrl":"https://doi.org/10.1142/s1793042124500581","url":null,"abstract":"<p>We fix a number field <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span> and study statistical properties of the ring <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>γ</mi><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">∩</mo><mi>K</mi></math></span><span></span> as <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>γ</mi></math></span><span></span> varies over algebraic numbers of a fixed degree <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>2</mn></math></span><span></span>. Given <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>≥</mo><mn>1</mn></math></span><span></span>, we explicitly compute the density of <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>γ</mi></math></span><span></span> for which <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>γ</mi><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">∩</mo><mi>K</mi><mo>=</mo><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy=\"false\">[</mo><mn>1</mn><mo stretchy=\"false\">/</mo><mi>k</mi><mo stretchy=\"false\">]</mo></math></span><span></span> and show that this does not depend on the number field <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span>. In particular, we show that the density of <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>γ</mi></math></span><span></span> for which <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>γ</mi><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">∩</mo><mi>K</mi><mo>=</mo><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span><span></span> is <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mfrac><mrow><mi>ζ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>ζ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></mrow></mfrac></math></span><span></span>. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, <i>Int. J. Number Theory</i> (2023), doi:10.1142/S1793042124500167], the authors define <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi><mo stretchy=\"false\">(</mo><mi>K</mi><mo>,</mo><mi>γ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> to be a certain finite subset of <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext>Spec</mtext></mst","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204336","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}

{"title":"Multi-partition analogue of q-binomial coefficients","authors":"Byungchan Kim, Hayan Nam, Myungjun Yu","doi":"10.1142/s1793042124500659","DOIUrl":"https://doi.org/10.1142/s1793042124500659","url":null,"abstract":"<p>We introduce the multi-Gaussian polynomial <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>M</mi><mo>,</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, a multi-partition analogue of the Gaussian polynomial (also known as <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>q</mi></math></span><span></span>-binomial coefficient), as the generating function for certain restricted multi-color partitions. We study basic properties of multi-Gaussian polynomials and non-symmetric properties of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>M</mi><mo>,</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. We also derive a Sylvester-type identity and its application.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204254","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}

{"title":"Corrigendum to “The discriminant of compositum of algebraic number fields”","authors":"Sudesh Kaur Khanduja","doi":"10.1142/s1793042124500489","DOIUrl":"https://doi.org/10.1142/s1793042124500489","url":null,"abstract":"<p>We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, <i>Int. J. Number Theory</i><b>15</b> (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204333","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}

{"title":"The Barnes–Hurwitz zeta cocycle at s = 0 and Ehrhart quasi-polynomials of triangles","authors":"Milton Espinoza","doi":"10.1142/s179304212450057x","DOIUrl":"https://doi.org/10.1142/s179304212450057x","url":null,"abstract":"<p>Following a theorem of Hayes, we give a geometric interpretation of the special value at <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi><mo>=</mo><mn>0</mn></math></span><span></span> of certain <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span>-cocycle on <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">PGL</mtext></mstyle></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℚ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi><mo>=</mo><mn>0</mn></math></span><span></span>, a generalization and a new proof of Hayes’ theorem, and an elegant summation formula for the zeroth coefficient of the Ehrhart quasi-polynomial of certain triangles in <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204340","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}

{"title":"Linear algebra and congruences for MacMahon’s k-rowed plane partitions","authors":"Shi-Chao Chen","doi":"10.1142/s1793042124500702","DOIUrl":"https://doi.org/10.1142/s1793042124500702","url":null,"abstract":"<p>In this paper, we provide an algorithm to detect linear congruences of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, the number of MacMahon’s <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span>-rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span>-rowed plane partition functions. We also show <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo stretchy=\"false\">)</mo></math></span><span></span> that the number of partitions at most <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi></math></span><span></span> parts always admits linear congruences.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204255","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}

{"title":"Fast computation of generalized dedekind sums","authors":"Preston Tranbarger, Jessica Wang","doi":"10.1142/s179304212450060x","DOIUrl":"https://doi.org/10.1142/s179304212450060x","url":null,"abstract":"<p>We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204260","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}

{"title":"The Manin–Peyre conjecture for certain multiprojective hypersurfaces","authors":"Xiaodong Zhao","doi":"10.1142/s1793042124500623","DOIUrl":"https://doi.org/10.1142/s1793042124500623","url":null,"abstract":"<p>By the circle method, an asymptotic formula is established for the number of integer points on certain hypersurfaces within multiprojective space. Using Möbius inversion and the modified hyperbola method, we prove the Manin–Peyre conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for certain smooth hypersurfaces in the multiprojective space of sufficiently large dimension.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204261","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}

{"title":"Higher Mertens constants for almost primes II","authors":"Jonathan Bayless, Paul Kinlaw, Jared Duker Lichtman","doi":"10.1142/s179304212450088x","DOIUrl":"https://doi.org/10.1142/s179304212450088x","url":null,"abstract":"<p>For <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>≥</mo><mn>1</mn></math></span><span></span>, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> denote the reciprocal sum up to <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span> of numbers with <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span> prime factors, counted with multiplicity. In prior work, the authors obtained estimates for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, extending Mertens’ second theorem, as well as a finer-scale estimate for <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> up to <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mo stretchy=\"false\">(</mo><mo>log</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">−</mo><mi>N</mi></mrow></msup></math></span><span></span> error for any <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi><mo>&gt;</mo><mn>0</mn></math></span><span></span>. In this paper, we establish the limiting behavior of the higher Mertens constants from the <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> estimate. We also extend these results to <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, and we comment on the general case <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>≥</mo><mn>4</mn></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204259","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}

{"title":"Variations on a theorem of Capelli","authors":"Pradipto Banerjee","doi":"10.1142/s1793042124500465","DOIUrl":"https://doi.org/10.1142/s1793042124500465","url":null,"abstract":"<p>Elementary irreducibility criteria are established for <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span> where <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>∈</mo><mi>ℤ</mi><mo stretchy=\"false\">[</mo><mi>x</mi><mo stretchy=\"false\">]</mo></math></span><span></span> is irreducible over <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℚ</mi></math></span><span></span> and <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span> is a prime. For instance, our main criterion implies that if <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span> is reducible over <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℚ</mi></math></span><span></span>, then <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> divides <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span> modulo <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span>. Among several applications, it is shown that if <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> has coefficients in <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">{</mo><mo stretchy=\"false\">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy=\"false\">}</mo></math></span><span></span>, then <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span> is irreducible over <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℚ</mi></math></span><span></span> excluding a couple of obvious exceptions. As another application, it is proved that if <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>&gt;</mo><mn>4</mn></math></span><span></span> and <span><math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204253","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}