When the Central Norm Equals 2 in the Simple Continued Fraction Expansion of a Quadratic Surd∗
نویسنده
چکیده
We complete the task, begun in [19], of determining when the central norm (determined by the infrastructure of the underlying real quadratic field) is equal to 2 in the simple continued fraction expansion of the associated quadratic surd.
منابع مشابه
On the real quadratic fields with certain continued fraction expansions and fundamental units
The purpose of this paper is to investigate the real quadratic number fields $Q(sqrt{d})$ which contain the specific form of the continued fractions expansions of integral basis element where $dequiv 2,3( mod 4)$ is a square free positive integer. Besides, the present paper deals with determining the fundamental unit$$epsilon _{d}=left(t_d+u_dsqrt{d}right) 2left.right > 1$$and $n_d$ and $m_d...
متن کاملPurely Periodic Nearest Square Continued Fractions
We give three sets of conditions to determine whether a real quadratic surd ξ = (P + √ D)/Q has a purely periodic nearest square continued fraction expansion. One set is a few inequalities involving only ξ and its conjugate ξ = (P − √ D)/Q. Another set is a few inequalities involving only P/ √ D and Q/ √ D. A third set of conditions and additional results are presented.
متن کاملQuadratic Irrational Integers with Partly Prescribed Continued Fraction Expansion
We generalise remarks of Euler and of Perron by explaining how to detail all quadratic integers for which the symmetric part of their continued fraction expansion commences with prescribed partial quotients. I last saw Bela Brindza, my once postdoctoral student, in April, 2002. I was working on the paper below and attempted to enthuse him with its results, particularly those concerning periodic...
متن کاملAll Solutions of the Diophantine Equation x 2 −
The main thrust of this article is to show how complete solutions of quadratic Diophantine equations can be given, for any positive discriminant, in terms of the continued fraction algorithm. This is in response to recent results by Zhang [4]-[6], wherein semi-simple continued fractions were introduced to generalize the well-known fact that solutions of quadratic Diophantine equations with valu...
متن کاملPell Equations: Non-Principal Lagrange Criteria and Central Norms
We provide a criterion for the central norm to be any value in the simple continued fraction expansion of √ D for any non-square integer D > 1. We also provide a simple criterion for the solvability of the Pell equation x2 − Dy2 = −1 in terms of congruence conditions modulo D.
متن کامل