Some formulas concerning the fundamental unit of a real quadratic field

The Unit Group of a Real Quadratic Field

While the unit group of an imaginary quadratic field is very simple, the unit group of a real quadratic field has nontrivial structure. Its study involves some geometry and analysis, but also it relates to Pell's equation and continued fractions, topics from elementary number theory.

Some Linear and Some Quadratic Recursion Formulas

§ 1. Introduction We shall mainly deal with linear recursion formulas of the type n-1 (l. 1) f(1) = 1 ; /(n) _ i~ ck f(n-k) (n = 2, 3,. . .), k=1 and with quadratic formulas of the type n-1 (1. 2) f(1) = 1 ; /(n) _ d, f(k)/(n-k) (n = 2,3. . .). k=1 We assume that ck > 0, d k > 0 (k = 1, 2,. . .). In a previous paper [1] we discussed (1. 1) under the condition X° ck = 1, and further special assu...

Some fundamental issues concerning degrees of unsolvability

Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT . The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is > 0 and < 0 ′. The second issue is to find a “smallness property” of an infinite, co-recursively enumerable set A ⊆ ω which ensures that the...

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...

On the modularity of elliptic curves over a composite field of some real quadratic fields