On a problem of Erdős and Graham

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• 作者：Szabolcs Tengely
• 关键词：Diophantine equations ; Blocks of consecutive integers ; Runge’s method
• 刊名：Periodica Mathematica Hungarica
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：72
• 期：1
• 页码：23-28
• 全文大小：364 KB
• 参考文献：1.M. Bauer, M.A. Bennett, On a question of Erdős and Graham. Enseign. Math. (2) 53(3–4), 259–264 (2007)MathSciNet MATH
  2.M.A. Bennett, N. Bruin, K. Győry, L. Hajdu, Powers from products of consecutive terms in arithmetic progression. Proc. Lond. Math. Soc. (3) 92(2), 273–306 (2006)MathSciNet CrossRef MATH
  3.M.A. Bennett, R. Van Luijk, Squares from blocks of consecutive integers: a problem of Erdős and Graham. Indag. Math. New Ser. 23(1–2), 123–127 (2012)CrossRef MATH
  4.L.E. Dickson, History of the Theory of Nnumbers. Vol II: Diophantine Analysis (Chelsea Publishing Co., New York, 1966)
  5.P. Erdős, Old and New Problems and Results in Combinatorial Number Theory (Université de Genéve, Geneva, 1980)MATH
  6.P. Erdős, Note on the product of consecutive integers (II). J. Lond. Math. Soc. 14, 245–249 (1939)MathSciNet CrossRef MATH
  7.P. Erdős, J.L. Selfridge, The product of consecutive integers is never a power. Ill. J. Math. 19, 292–301 (1975)MathSciNet MATH
  8.A. Grytczuk, A. Schinzel, in On Runge’s Theorem About Diophantine Equations. Sets, Graphs and Numbers. A Birthday Salute to Vera T. Sós and András Hajnal (North-Holland Publishing Company, Amsterdam, 1992), pp. 329–356
  9.K. Győry, On the diophantine equation \(n(n+1)\dots (n+k-1)=bx^\ell \) . Acta Arith. 83(1), 87–92 (1998)MathSciNet MATH
  10.K. Győry, L. Hajdu, N. Saradha, On the Diophantine equation \(n(n+d)\cdots (n+(k-1)d)=by^l\) . Can. Math. Bull. 47(3), 373–388 (2004)MathSciNet CrossRef MATH
  11.L. Hajdu, Sz Tengely, R. Tijdeman, Cubes in products of terms in arithmetic progression. Publ. Math. Debr. 74(1–2), 215–232 (2009)MathSciNet MATH
  12.D.L. Hilliker, E.G. Straus, Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theorem. Trans. Am. Math. Soc. 280(2), 637–657 (1983)MathSciNet MATH
  13.N. Hirata-Kohno, S. Laishram, T.N. Shorey, R. Tijdeman, An extension of a theorem of Euler. Acta Arith. 129(1), 71–102 (2007)MathSciNet CrossRef MATH
  14.S. Laishram, T.N. Shorey, The equation \(n(n+d)\cdots (n+(k-1)d)=by^2\) with \(\omega (d)\le 6\) or \(d\le 10^{10}\) . Acta Arith. 129(3), 249–305 (2007)MathSciNet CrossRef MATH
  15.F. Luca, P.G. Walsh, On a diophantine equation related to a conjecture of Erdős and Graham. Glas. Mat. Ser. III 42(2), 281–289 (2007)MathSciNet CrossRef MATH
  16.R. Obláth, Über das Produkt fünf aufeinander folgender Zahlen in einer arithmetischen Reihe. Publ. Math. Debr. 1, 222–226 (1950)MATH
  17.O. Rigge, Über ein diophantisches problem. In 9th Congress Math. Scand., pp. 155–160; Mercator 1939, Helsingfors (1938)
  18.C. Runge, Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen. J. Reine Angew. Math. 100, 425–435 (1887)MathSciNet MATH
  19.A. Sankaranarayanan, N. Saradha, Estimates for the solutions of certain Diophantine equations by Runge’s method. Int. J. Number Theory 4(3), 475–493 (2008)MathSciNet CrossRef MATH
  20.N. Saradha, On perfect powers in products with terms from arithmetic progressions. Acta Arith. 82(2), 147–172 (1997)MathSciNet MATH
  21.N. Saradha, T.N. Shorey, Almost squares in arithmetic progression. Compos. Math. 138(1), 73–111 (2003)MathSciNet CrossRef MATH
  22.A. Schinzel, An improvement of Runge’s theorem on Diophantine equations. Comment. Pontif. Acad. Sci. 2(20), 1–9 (1969)MathSciNet MATH
  23.M. Skałba, Products of disjoint blocks of consecutive integers which are powers. Colloq. Math. 98(1), 1–3 (2003)MathSciNet CrossRef MATH
  24.N.P. Smart, The Algorithmic Resolution of Diophantine Equations, London Mathematical Society Student Texts, vol. 41 (Cambridge University Press, Cambridge, 1998)CrossRef
  25.W. A. Stein et al. Sage Mathematics Software (Version 6.3). The Sage Development Team, http://​www.​sagemath.​org (2014)
  26.Sz Tengely, On the Diophantine equation \(F(x)=G(y)\) . Acta Arith. 110(2), 185–200 (2003)MathSciNet CrossRef MATH
  27.M. Ulas, On products of disjoint blocks of consecutive integers. Enseign. Math. 51(3–4), 331–334 (2005)MathSciNet MATH
  28.P.G. Walsh, A quantitative version of Runge’s theorem on Diophantine equations. Acta Arith. 62(2), 157–172 (1992)MathSciNet MATH
  
• 作者单位：Szabolcs Tengely (1)
  
  1. Mathematical Institute, University of Derecen, P.O.Box 12, Debrecen, 4010, Hungary
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2829

文摘

In this paper we provide bounds for the size of the solutions of the Diophantine equation $$\begin{aligned} x(x+1)(x+2)(x+3)(x+m)(x+m+1)(x+m+2)(x+m+3)=y^2, \end{aligned}$$where \(4\le m\in \mathbb {N}\) is a parameter. We also determine all integral solutions for \(1\le m\le 10^6.\) Keywords Diophantine equations Blocks of consecutive integers Runge’s method Mathematics Subject Classification Primary 11D61 Secondary 11Y50 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (28) References1.M. Bauer, M.A. Bennett, On a question of Erdős and Graham. Enseign. Math. (2) 53(3–4), 259–264 (2007)MathSciNetMATH2.M.A. Bennett, N. Bruin, K. Győry, L. Hajdu, Powers from products of consecutive terms in arithmetic progression. Proc. Lond. Math. Soc. (3) 92(2), 273–306 (2006)MathSciNetCrossRefMATH3.M.A. Bennett, R. Van Luijk, Squares from blocks of consecutive integers: a problem of Erdős and Graham. Indag. Math. New Ser. 23(1–2), 123–127 (2012)CrossRefMATH4.L.E. Dickson, History of the Theory of Nnumbers. Vol II: Diophantine Analysis (Chelsea Publishing Co., New York, 1966)5.P. Erdős, Old and New Problems and Results in Combinatorial Number Theory (Université de Genéve, Geneva, 1980)MATH6.P. Erdős, Note on the product of consecutive integers (II). J. Lond. Math. Soc. 14, 245–249 (1939)MathSciNetCrossRefMATH7.P. Erdős, J.L. Selfridge, The product of consecutive integers is never a power. Ill. J. Math. 19, 292–301 (1975)MathSciNetMATH8.A. Grytczuk, A. Schinzel, in On Runge’s Theorem About Diophantine Equations. Sets, Graphs and Numbers. A Birthday Salute to Vera T. Sós and András Hajnal (North-Holland Publishing Company, Amsterdam, 1992), pp. 329–3569.K. Győry, On the diophantine equation \(n(n+1)\dots (n+k-1)=bx^\ell \). Acta Arith. 83(1), 87–92 (1998)MathSciNetMATH10.K. Győry, L. Hajdu, N. Saradha, On the Diophantine equation \(n(n+d)\cdots (n+(k-1)d)=by^l\). Can. Math. Bull. 47(3), 373–388 (2004)MathSciNetCrossRefMATH11.L. Hajdu, Sz Tengely, R. Tijdeman, Cubes in products of terms in arithmetic progression. Publ. Math. Debr. 74(1–2), 215–232 (2009)MathSciNetMATH12.D.L. Hilliker, E.G. Straus, Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theorem. Trans. Am. Math. Soc. 280(2), 637–657 (1983)MathSciNetMATH13.N. Hirata-Kohno, S. Laishram, T.N. Shorey, R. Tijdeman, An extension of a theorem of Euler. Acta Arith. 129(1), 71–102 (2007)MathSciNetCrossRefMATH14.S. Laishram, T.N. Shorey, The equation \(n(n+d)\cdots (n+(k-1)d)=by^2\) with \(\omega (d)\le 6\) or \(d\le 10^{10}\). Acta Arith. 129(3), 249–305 (2007)MathSciNetCrossRefMATH15.F. Luca, P.G. Walsh, On a diophantine equation related to a conjecture of Erdős and Graham. Glas. Mat. Ser. III 42(2), 281–289 (2007)MathSciNetCrossRefMATH16.R. Obláth, Über das Produkt fünf aufeinander folgender Zahlen in einer arithmetischen Reihe. Publ. Math. Debr. 1, 222–226 (1950)MATH17.O. Rigge, Über ein diophantisches problem. In 9th Congress Math. Scand., pp. 155–160; Mercator 1939, Helsingfors (1938)18.C. Runge, Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen. J. Reine Angew. Math. 100, 425–435 (1887)MathSciNetMATH19.A. Sankaranarayanan, N. Saradha, Estimates for the solutions of certain Diophantine equations by Runge’s method. Int. J. Number Theory 4(3), 475–493 (2008)MathSciNetCrossRefMATH20.N. Saradha, On perfect powers in products with terms from arithmetic progressions. Acta Arith. 82(2), 147–172 (1997)MathSciNetMATH21.N. Saradha, T.N. Shorey, Almost squares in arithmetic progression. Compos. Math. 138(1), 73–111 (2003)MathSciNetCrossRefMATH22.A. Schinzel, An improvement of Runge’s theorem on Diophantine equations. Comment. Pontif. Acad. Sci. 2(20), 1–9 (1969)MathSciNetMATH23.M. Skałba, Products of disjoint blocks of consecutive integers which are powers. Colloq. Math. 98(1), 1–3 (2003)MathSciNetCrossRefMATH24.N.P. Smart, The Algorithmic Resolution of Diophantine Equations, London Mathematical Society Student Texts, vol. 41 (Cambridge University Press, Cambridge, 1998)CrossRef25.W. A. Stein et al. Sage Mathematics Software (Version 6.3). The Sage Development Team, http://​www.​sagemath.​org (2014)26.Sz Tengely, On the Diophantine equation \(F(x)=G(y)\). Acta Arith. 110(2), 185–200 (2003)MathSciNetCrossRefMATH27.M. Ulas, On products of disjoint blocks of consecutive integers. Enseign. Math. 51(3–4), 331–334 (2005)MathSciNetMATH28.P.G. Walsh, A quantitative version of Runge’s theorem on Diophantine equations. Acta Arith. 62(2), 157–172 (1992)MathSciNetMATH About this Article Title On a problem of Erdős and Graham Journal Periodica Mathematica Hungarica Volume 72, Issue 1 , pp 23-28 Cover Date2016-03 DOI 10.1007/s10998-015-0098-8 Print ISSN 0031-5303 Online ISSN 1588-2829 Publisher Springer Netherlands Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords Diophantine equations Blocks of consecutive integers Runge’s method Primary 11D61 Secondary 11Y50 Authors Szabolcs Tengely (1) Author Affiliations 1. Mathematical Institute, University of Derecen, P.O.Box 12, Debrecen, 4010, Hungary Continue reading... To view the rest of this content please follow the download PDF link above.