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Rational and algebraic approximations of algebraic numbers and their application 被引量:3

Rational and algebraic approximations of algebraic numbers and their application
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摘要 Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel's method.As an application of this result,it is proved that; if D>0 is not a square,and ε=x0 +y0 D denotes the fundamental solution of x2-Dy2=-1,then x2+1=Dy4 is solvable if and only if y0=A2 where A is an integer.Moreover,if ≥64,then x2+1=Dy4 has at most one positive integral solution (x,y). Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method.As an application of this result,it is proved that; if D>0 is not a square,and ε=x0 +y0 D denotes the fundamental solution of x2-Dy2=-1,then x2+1=Dy4 is solvable if and only if y0=A2 where A is an integer.Moreover,if ≥64,then x2+1=Dy4 has at most one positive integral solution (x,y).
作者 袁平之
出处 《Science China Mathematics》 SCIE 1997年第10期1045-1051,共7页 中国科学:数学(英文版)
基金 Project supported by the National Natural Science Foundation of China.
关键词 HYPERGEOMETRIC functions DIOPHANTINE APPROXIMATIONS imaginary QUADRATIC fields QUARTIC equations. hypergeometric functions,diophantine approximations,imaginary quadratic fields,quartic equations.
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  • 1YUAN PingZhi 1 & ZHANG ZhongFeng 2 1 School of Mathematics,South China Normal University,Guangzhou 510631,China,2 School of Mathematics & Computational Science,Sun Yat-Sen University,Guangzhou 510275,China.On the diophantine equation X^2-(1+a^2)Y^4 =-2a[J].Science China Mathematics,2010,53(8):2143-2158. 被引量:7
  • 2Pingzhi Yuan.Rational and algebraic approximations of algebraic numbers and their application[J]. Science in China Series A: Mathematics . 1997 (10)
  • 3He B,Togbe A,Walsh P G.On the Diophantine equation x 2 - (2 2m + 1)y 4 = -2 2m. Publications Mathematicae Debrecen . 2008
  • 4Luca F,Walsh P G.Squares in Lehmer sequences and some Diophantine equations. Acta Arithmetica . 2001
  • 5Tengely S.Effective Methods for Diophantine Equations. . 2005
  • 6Voutier P M.Thue’s fundamentaltheorem I: the general case. Acta Arithmetica . 2010
  • 7Walsh P G.On the number of large integer points on elliptic curves. Acta Arithmetica . 2009
  • 8Akhtari S.The method of Thue and Siegel for binary quartic forms. Acta Arithmetica . 2010
  • 9Chen J H,Voutier P M.A complete solution of the Diophantine equation x 2 + 1 = dy 4 and a related family of quartic Thue equations. Journal of Number Theory . 1997
  • 10Dujella A.Continued fractions and RSA with small secret exponent. Tatra Mt Math Publ . 2004

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