International Journal of Number Theory最新文献

{"title":"On the solutions of some Lebesgue–Ramanujan–Nagell type equations","authors":"Elif Kızıldere Mutlu, Gökhan Soydan","doi":"10.1142/s1793042124500593","DOIUrl":"https://doi.org/10.1142/s1793042124500593","url":null,"abstract":"<p>Denote by <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi><mo>=</mo><mi>h</mi><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">−</mo><mi>p</mi><mo stretchy=\"false\">)</mo></math></span><span></span> the class number of the imaginary quadratic field <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℚ</mi><mo stretchy=\"false\">(</mo><msqrt><mrow><mo stretchy=\"false\">−</mo><mi>p</mi></mrow></msqrt><mo stretchy=\"false\">)</mo></math></span><span></span> with <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span> prime. It is well known that <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi><mo>=</mo><mn>1</mn></math></span><span></span> for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><mo>∈</mo><mo stretchy=\"false\">{</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>1</mn><mn>1</mn><mo>,</mo><mn>1</mn><mn>9</mn><mo>,</mo><mn>4</mn><mn>3</mn><mo>,</mo><mn>6</mn><mn>7</mn><mo>,</mo><mn>1</mn><mn>6</mn><mn>3</mn><mo stretchy=\"false\">}</mo></math></span><span></span>. Recently, all the solutions of the Diophantine equation <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><mn>4</mn><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> with <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi><mo>=</mo><mn>1</mn></math></span><span></span> were given by Chakraborty <i>et al</i>. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, <i>Publ. Math. Debrecen</i><b>97</b>(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> in unknown integers <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo>,</mo></math></span><span></span> where <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi><mo>≥</mo><mn>0</mn></math></span><span></span>, <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>r</mi><mo>≥</mo><mn>3</mn></math></span><span></span>, <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>3</mn></math></span><span></span>, <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi><mo>∈</mo><mo st","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599270","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 non-vanishing of Fourier coefficients of half-integral weight cuspforms","authors":"Jun-Hwi Min","doi":"10.1142/s1793042124500805","DOIUrl":"https://doi.org/10.1142/s1793042124500805","url":null,"abstract":"<p>We prove the best possible upper bounds of the gaps between non-vanishing Fourier coefficients of half-integral weight cuspforms. This improves the works of Balog–Ono and Thorner. We also show an asymptotic formula of central modular <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi></math></span><span></span>-values for short intervals.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599461","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":"Congruence classes for modular forms over small sets","authors":"Subham Bhakta, Srilakshmi Krishnamoorthy, R. Muneeswaran","doi":"10.1142/s1793042124500799","DOIUrl":"https://doi.org/10.1142/s1793042124500799","url":null,"abstract":"<p>Serre showed that for any integer <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi><mo>,</mo><mspace width=\"0.25em\"></mspace><mi>a</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo>≡</mo><mn>0</mn><mspace width=\"0.3em\"></mspace><mo stretchy=\"false\">(</mo><mo>mod</mo><mspace width=\"0.3em\"></mspace><mi>m</mi><mo stretchy=\"false\">)</mo></math></span><span></span> for almost all <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>,</mo></math></span><span></span> where <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is the <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mstyle><mtext>th</mtext></mstyle></math></span><span></span> Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>#</mi><msub><mrow><mo stretchy=\"false\">{</mo><mi>a</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mspace width=\"0.3em\"></mspace><mo stretchy=\"false\">(</mo><mo>mod</mo><mspace width=\"0.3em\"></mspace><mi>m</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">}</mo></mrow><mrow><mi>n</mi><mo>≤</mo><mi>x</mi></mrow></msub></math></span><span></span> over the set of integers with <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span><span></span> many prime factors. Moreover, we show that any residue class <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy=\"false\">/</mo><mi>m</mi><mi>ℤ</mi></math></span><span></span> can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi><mo>.</mo></math></span><span></span></p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599462","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":"Moments of Dirichlet L-functions to a fixed modulus over function fields","authors":"Peng Gao, Liangyi Zhao","doi":"10.1142/s1793042124500738","DOIUrl":"https://doi.org/10.1142/s1793042124500738","url":null,"abstract":"<p>In this paper, we establish the expected order of magnitude of the <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span>th-moment of central values of the family of Dirichlet <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi></math></span><span></span>-functions to a fixed prime modulus over function fields for all real <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>≥</mo><mn>0</mn></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599262","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":"Near-squares in binary recurrence sequences","authors":"Nikos Tzanakis, Paul Voutier","doi":"10.1142/s1793042124500787","DOIUrl":"https://doi.org/10.1142/s1793042124500787","url":null,"abstract":"<p>We call an integer a <i>near-square</i> if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>≥</mo><mn>3</mn></math></span><span></span> by <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></math></span><span></span>, <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn></math></span><span></span> and <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi><mo stretchy=\"false\">+</mo><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>a</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo></math></span><span></span> for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>0</mn></math></span><span></span>. We show that for a given <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>≥</mo><mn>3</mn></math></span><span></span>, there is at most one <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>5</mn></math></span><span></span> such that <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is a near-square. With the exceptions of <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>u</mi></mrow><mrow><mn>6</mn></mrow></msub><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span> and <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>u</mi></mrow><mrow><mn>7</mn></mrow></msub><mo stretchy=\"false\">(</mo><mn>6</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn><mn>3</mn><mn>9</mn><mo stretchy=\"false\">⋅</mo><mn>1</mn><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span>, any such <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo></math></span><span></span> can be a nea","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":"140599447","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 cuspidal cohomology of GL3/ℚ and cubic fields","authors":"Avner Ash, Dan Yasaki","doi":"10.1142/s1793042124500829","DOIUrl":"https://doi.org/10.1142/s1793042124500829","url":null,"abstract":"<p>We investigate the subspace of the homology of a congruence subgroup <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"normal\">Γ</mi></math></span><span></span> of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">SL</mtext></mstyle></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℤ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> with coefficients in the Steinberg module <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">St</mtext></mstyle><mo stretchy=\"false\">(</mo><msup><mrow><mi>ℚ</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span> which is spanned by certain modular symbols formed using the units of a totally real cubic field <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>E</mi></math></span><span></span>. By Borel–Serre duality, <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi mathvariant=\"normal\">Γ</mi><mo>,</mo><mstyle><mtext mathvariant=\"normal\">St</mtext></mstyle><mo stretchy=\"false\">(</mo><msup><mrow><mi>ℚ</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span> is isomorphic to <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi mathvariant=\"normal\">Γ</mi><mo>,</mo><mi>ℚ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. The Borel–Serre duals of the modular symbols in question necessarily lie in the cuspidal cohomology <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi>H</mi></mrow><mrow><mstyle><mtext mathvariant=\"normal\">cusp</mtext></mstyle></mrow><mrow><mn>3</mn></mrow></msubsup><mo stretchy=\"false\">(</mo><mi mathvariant=\"normal\">Γ</mi><mo>,</mo><mi>ℚ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. Their span is a naturally defined subspace <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi><mo stretchy=\"false\">(</mo><mi mathvariant=\"normal\">Γ</mi><mo>,</mo><mi>E</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi>H</mi></mrow><mrow><mstyle><mtext mathvariant=\"normal\">cusp</mtext></mstyle></mrow><mrow><mn>3</mn></mrow></msubsup><mo stretchy=\"false\">(</mo><mi mathvariant=\"normal\">Γ</mi><mo>,</mo><mi>ℚ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. Using a computer, we study where <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi><mo stretchy=\"false\">(</mo><mi mathvariant=\"normal\">Γ</mi><mo>,</mo><mi>E</mi><mo stretchy=\"false\">)</mo></math></span><span></span> sits between <span>","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":"140599115","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":"Dynatomic Galois groups for a family of quadratic rational maps","authors":"David Krumm, Allan Lacy","doi":"10.1142/s1793042124500830","DOIUrl":"https://doi.org/10.1142/s1793042124500830","url":null,"abstract":"<p>For every nonconstant rational function <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>ϕ</mi><mo>∈</mo><mi>ℚ</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, the Galois groups of the dynatomic polynomials of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>ϕ</mi></math></span><span></span> encode various properties of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>ϕ</mi></math></span><span></span> are of interest in the subject of arithmetic dynamics. We study here the structure of these Galois groups as <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>ϕ</mi></math></span><span></span> varies in a particular one-parameter family of maps, namely, the quadratic rational maps having a critical point of period 2. In particular, we provide explicit descriptions of the third and fourth dynatomic Galois groups for maps in this family.</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":"140599013","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":"Relations of multiple t-values of general level","authors":"Zhonghua Li, Zhenlu Wang","doi":"10.1142/s1793042124500696","DOIUrl":"https://doi.org/10.1142/s1793042124500696","url":null,"abstract":"<p>We study the relations of multiple <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>-values of general level. The generating function of sums of multiple <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>-(star) values of level <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi></math></span><span></span> with fixed weight, depth and height is represented by the generalized hypergeometric function <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span><span></span>, which generalizes the results for multiple zeta(-star) values and multiple <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>-(star) values. As applications, we obtain formulas for the generating functions of sums of multiple <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>-(star) values of level <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi></math></span><span></span> with height one and maximal height and a weighted sum formula for sums of multiple <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>-(star) values of level <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi></math></span><span></span> with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman’s restricted sum formulas for multiple <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>-(star) values of level <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi></math></span><span></span>. Some evaluations of multiple <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>-star values of level <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span> with one–two–three indices are given.</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":"140599091","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":"Rational points on x3 + x2y2 + y3 = k","authors":"Xiaoan Lang, Jeremy Rouse","doi":"10.1142/s1793042124500878","DOIUrl":"https://doi.org/10.1142/s1793042124500878","url":null,"abstract":"<p>We study the problem of determining, given an integer <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span>, the rational solutions to <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>:</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>z</mi><mspace width=\".17em\"></mspace><mo stretchy=\"false\">+</mo><mspace width=\".17em\"></mspace><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace width=\".17em\"></mspace><mo stretchy=\"false\">+</mo><mspace width=\".17em\"></mspace><msup><mrow><mi>y</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>z</mi><mo>=</mo><mi>k</mi><msup><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span><span></span>. For <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>≠</mo><mn>0</mn></math></span><span></span>, the curve <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span><span></span> has genus <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span> and its Jacobian is isogenous to the product of three elliptic curves <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>k</mi></mrow></msub></math></span><span></span>, <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>k</mi></mrow></msub></math></span><span></span>, <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>E</mi></mrow><mrow><mn>3</mn><mo>,</mo><mi>k</mi></mrow></msub></math></span><span></span>. We explicitly determine the rational points on <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span><span></span> under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>∈</mo><mi>ℚ</mi></math></span><span></span>.</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":"140599095","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":"Irreducibility and galois groups of truncated binomial polynomials","authors":"Shanta Laishram, Prabhakar Yadav","doi":"10.1142/s1793042124500817","DOIUrl":"https://doi.org/10.1142/s1793042124500817","url":null,"abstract":"<p>For positive integers <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mi>m</mi></math></span><span></span>, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>:</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced close=\")\" open=\"(\" separators=\"\"><mfrac linethickness=\"0\"><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></mfrac></mfenced><msup><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msup><mo>=</mo><mfenced close=\")\" open=\"(\" separators=\"\"><mfrac linethickness=\"0\"><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></mfrac></mfenced><mo stretchy=\"false\">+</mo><mfenced close=\")\" open=\"(\" separators=\"\"><mfrac linethickness=\"0\"><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></mfrac></mfenced><mi>x</mi><mo stretchy=\"false\">+</mo><mo>…</mo><mo stretchy=\"false\">+</mo><mfenced close=\")\" open=\"(\" separators=\"\"><mfrac linethickness=\"0\"><mrow><mi>n</mi></mrow><mrow><mi>m</mi></mrow></mfrac></mfenced><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span><span></span> be the truncated binomial expansion of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">+</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> consisting of all terms of degree <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo>≤</mo><mi>m</mi><mo>.</mo></math></span><span></span> It is conjectured that for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>&gt;</mo><mi>m</mi><mo stretchy=\"false\">+</mo><mn>1</mn></math></span><span></span>, the polynomial <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is irreducible. We confirm this conjecture when <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mi>m</mi><mo>≤</mo><mi>n</mi><mo>&lt;</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>m</mi><mo stretchy=\"false\">+</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mn>1</mn><mn>0</mn></mrow></msup><mo>.</mo></math></span><span></span> Also we show for any <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>r</mi><mo>≥</mo><mn>1</mn><mn>0</mn></math></span><span></span> and <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mi>m</mi><mo>≤</mo><mi>n</mi><mo>&lt;</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>m</mi><mo stretchy=\"false\">+</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>r</mi><mo stretchy=\"false\">+</mo><mn>1</mn>","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":"140599248","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}