{"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":"57 1","pages":""},"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":"90 1","pages":""},"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":"53 1","pages":""},"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":"49 1","pages":""},"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":"14 1","pages":""},"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>></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><</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><</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":"1 1","pages":""},"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}
{"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":"6 1","pages":""},"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":"32 1","pages":""},"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":"39 1","pages":""},"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":"141 1","pages":""},"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}
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