International Journal of Number Theory最新文献

{"title":"Finite sequences of integers expressible as sums of two squares","authors":"Ajai Choudhry, Bibekananda Maji","doi":"10.1142/s1793042124500866","DOIUrl":"https://doi.org/10.1142/s1793042124500866","url":null,"abstract":"<p>This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> such that <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mi>h</mi></math></span><span></span> and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo stretchy=\"false\">+</mo><mi>k</mi></math></span><span></span> are all sums of two squares where <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi></math></span><span></span> and <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span> are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> in parametric terms such that all the four integers <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>4</mn></math></span><span></span> are sums of two squares. We also find infinitely many integers <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> such that all the five integers <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>4</mn><mo>,</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>5</mn></math></span><span></span> are sums of two squares, and finally, we find infinitely many arithmetic progressions, with common difference <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mn>4</mn></math></span><span></span>, of five integers all of which are sums of two squares.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599249","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 genus of a quotient of several types of numerical semigroups","authors":"Kyeongjun Lee, Hayan Nam","doi":"10.1142/s1793042124500891","DOIUrl":"https://doi.org/10.1142/s1793042124500891","url":null,"abstract":"<p>Finding the Frobenius number and the genus of any numerical semigroup <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi></math></span><span></span> is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140602443","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":"Some separable integer partition classes","authors":"Y. H. Chen, Thomas Y. He, F. Tang, J. J. Wei","doi":"10.1142/s1793042124500660","DOIUrl":"https://doi.org/10.1142/s1793042124500660","url":null,"abstract":"<p>Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span>. We also extend separable integer partition classes with modulus <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325648","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":"Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues","authors":"Jiseong Kim","doi":"10.1142/s1793042124500775","DOIUrl":"https://doi.org/10.1142/s1793042124500775","url":null,"abstract":"<p>By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><mi>L</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo>,</mo><mi>ℤ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><mi>L</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo>,</mo><mi>ℤ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. Furthermore, we present a conditional result regarding sign changes of these coefficients.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325498","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":"Variance of primes in short residue classes for function fields","authors":"Stephan Baier, Arkaprava Bhandari","doi":"10.1142/s1793042124500763","DOIUrl":"https://doi.org/10.1142/s1793042124500763","url":null,"abstract":"<p>Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, <i>Int. Math. Res. Not.</i><b>2014</b>(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>Q</mi></math></span><span></span> is a polynomial in <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>T</mi><mo stretchy=\"false\">]</mo></math></span><span></span> such that <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>Q</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>≠</mo><mn>0</mn></math></span><span></span>. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325446","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":"Multiplier systems for Siegel modular groups","authors":"Eberhard Freitag, Adrian Hauffe-Waschbüsch","doi":"10.1142/s1793042124500684","DOIUrl":"https://doi.org/10.1142/s1793042124500684","url":null,"abstract":"<p>Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, <i>C. R. Acad. Sci. Paris</i><b>287</b> (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in <i>Automorphic Forms and Representations of Algebraic Groups Over Local Fields</i>, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>g</mi><mo>&gt;</mo><mn>1</mn></math></span><span></span> must be integral or half integral. Actually he proved that for a system <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>v</mi><mo stretchy=\"false\">(</mo><mi>M</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of complex numbers of absolute value 1</p><p><span><math altimg=\"eq-00003.gif\" display=\"block\" overflow=\"scroll\"><mtable columnalign=\"left\"><mtr><mtd columnalign=\"right\"><mspace width=\"8.5pc\"></mspace><mi>v</mi><mo stretchy=\"false\">(</mo><mi>M</mi><mo stretchy=\"false\">)</mo><mo>det</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>C</mi><mi>Z</mi><mo stretchy=\"false\">+</mo><mi>D</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>r</mi><mo>∈</mo><mi>ℝ</mi><mo stretchy=\"false\">)</mo><mspace width=\"8.5pc\"></mspace><mo stretchy=\"false\">(</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mtd><mtd></mtd></mtr></mtable></math></span><span></span></p><p>can be an automorphy factor only if <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mi>r</mi></math></span><span></span> is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, <i>Math. Ann.</i><b>159</b> (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace width=\".275em\"></mspace><mo stretchy=\"false\">(</mo><mi>n</mi><mo>≥</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mspace width=\".275em\"></mspace><mo stretchy=\"false\">(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math></span><span></span>, <i>Publ. Math. Inst. Hautes Études Sci.</i><b>33</b> (1967) 59–137] of Bass–Milnor–Serre.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325840","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 minimal odd excludant and Euler’s partition theorem","authors":"Andrew Y. Z. Wang, Zheng Xu","doi":"10.1142/s1793042124500714","DOIUrl":"https://doi.org/10.1142/s1793042124500714","url":null,"abstract":"<p>In this work, we establish two interesting partition identities involving the minimal odd excludant, which has attracted great attention in recent years. In particular, we find a strong refinement of Euler’s celebrated theorem that the number of partitions of an integer into odd parts equals the number of partitions of that integer into distinct parts.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325447","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":"Reciprocity formulae for generalized Dedekind–Rademacher sums attached to three Dirichlet characters and related polynomial reciprocity formulae","authors":"Brad Isaacson","doi":"10.1142/s1793042124500726","DOIUrl":"https://doi.org/10.1142/s1793042124500726","url":null,"abstract":"<p>We define a three-character analogue of the generalized Dedekind–Rademacher sum introduced by Hall, Wilson, and Zagier and prove its reciprocity formula which contains all of the reciprocity formulas in the literature for generalized Dedekind–Rademacher sums attached (and not attached) to Dirichlet characters as special cases. Additionally, we prove related polynomial reciprocity formulas which contain all of the polynomial reciprocity formulas in the literature as special cases, such as those given by Carlitz, Beck &amp; Kohl, and the present author.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325449","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":"On almost-prime k-tuples","authors":"Bin Chen","doi":"10.1142/s1793042124500751","DOIUrl":"https://doi.org/10.1142/s1793042124500751","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>τ</mi></math></span><span></span> denote the divisor function and <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">ℋ</mi><mo>=</mo><mo stretchy=\"false\">{</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">}</mo></math></span><span></span> be an admissible set. We prove that there are infinitely many <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> for which the product <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> is square-free and <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mi>τ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo>≤</mo><mo stretchy=\"false\">⌊</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">⌋</mo></math></span><span></span>, where <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span><span></span> is asymptotic to <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mfrac><mrow><mn>2</mn><mn>1</mn><mn>2</mn><mn>6</mn></mrow><mrow><mn>2</mn><mn>8</mn><mn>5</mn><mn>3</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span>. It improves a previous result of Ram Murty and Vatwani, replacing <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn><mo stretchy=\"false\">/</mo><mn>4</mn></math></span><span></span> by <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mn>1</mn><mn>2</mn><mn>6</mn><mo stretchy=\"false\">/</mo><mn>2</mn><mn>8</mn><mn>5</mn><mn>3</mn></math></span><span></span>. The main ingredients in our proof are the higher rank Selberg sieve and Irving–Wu–Xi estimate for the divisor function in arithmetic progressions to smooth moduli.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325724","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":"On the Diophantine equation σ2(X¯n) = σn(X¯n)","authors":"Piotr Miska, Maciej Ulas","doi":"10.1142/s1793042124500635","DOIUrl":"https://doi.org/10.1142/s1793042124500635","url":null,"abstract":"<p>In this paper, we investigate the set <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of positive integer solutions of the title Diophantine equation. In particular, for a given <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> we prove boundedness of the number of solutions, give precise upper bound on the common value of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mover accent=\"false\"><mrow><mi>X</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mover accent=\"false\"><mrow><mi>X</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> together with the biggest value of the variable <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> appearing in the solution. Moreover, we enumerate all solutions for <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≤</mo><mn>1</mn><mn>6</mn></math></span><span></span> and discuss the set of values of <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">/</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo stretchy=\"false\">−</mo><mn>1</mn></mrow></msub></math></span><span></span> over elements of <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325497","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}