scholarly journals POLYNOMIAL SOLUTIONS OF PELL'S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS

2003 ◽  
Vol 67 (01) ◽  
pp. 16-28 ◽  
Author(s):  
J. MCLAUGHLIN
Keyword(s):  
Quadratic Fields ◽  
Pell’S Equation ◽  
Real Quadratic Fields ◽  
Polynomial Solutions ◽  
Fundamental Units ◽  
Real Quadratic

scholarly journals Units of Real Quadratic Fields

1971 ◽  
Vol 44 ◽  
pp. 51-55 ◽  
Author(s):  
Akira Takaku
Keyword(s):  
Positive Solution ◽  
Quadratic Fields ◽  
Pell’S Equation ◽  
Real Quadratic Fields ◽  
Real Quadratic

1. Let D be a positive square-free integer. Throughout this note we shall use the following notations;d = d(D): the discriminant of ,t0, u0: the least positive solution of Pell’s equation t2 — du2 = 4,

scholarly journals Lower bounds for fundamental units of real quadratic fields

2002 ◽  
Vol 166 ◽  
pp. 29-37 ◽  
Author(s):  
Koshi Tomita ◽  
Kouji Yamamuro
Keyword(s):  
Lower Bounds ◽  
Continued Fraction ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Integral Basis ◽  
Continued Fraction Expansion ◽  
Real Quadratic Fields ◽  
Fundamental Units ◽  
Real Quadratic

AbstractLet d be a square-free positive integer and l(d) be the period length of the simple continued fraction expansion of ωd, where ωd is integral basis of ℤ[]. Let εd = (td + ud)/2 (> 1) be the fundamental unit of the real quadratic field ℚ(). In this paper new lower bounds for εd, td, and ud are described in terms of l(d). The lower bounds of εd are sharper than the known bounds and those of td and ud have been yet unknown. In order to show the strength of the method of the proof, some interesting examples of d are given for which εd and Yokoi’s d-invariants are determined explicitly in relation to continued fractions of the form .

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