scholarly journals Explicit representation of fundamental units of some quadratic fields

1995 ◽  
Vol 71 (2) ◽  
pp. 41-43 ◽  
Author(s):  
Koshi Tomita
Keyword(s):  
Explicit Representation ◽  
Quadratic Fields ◽  
Fundamental Units

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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