Expansions of quadratic maps in prime fields

Expansions of quadratic maps in prime fields

Let f(x) = ax+bx+c ∈ Z[x] be a quadratic polynomial with a ̸≡ 0 mod p. Take z ∈ Fp and let Oz = {fi(z)}i∈Z+ be the orbit of z under f , where fi(z) = f(fi−1(z)) and f0(z) = z. For M < |Oz|, We study the diameter of the partial orbit OM = {z, f(z), f2(z), . . . , fM−1(z)} and prove that there exists c1 > 0 such that diam OM & min { Mp c1 , 1 log p M 4 5p 1 5 ,M 1 13 log logM } . For a complete or...

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...

The Distribution of Prime Ideals of Imaginary Quadratic Fields

Irregularity of Prime Numbers over Real Quadratic Fields

The concept of regular and irregular primes has played an important role in number theory at least since the time of Kummer. We extend this concept to the setting of arbitrary totally real number fields k0, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. Once we have defined k0-regular primes and the index of k0-irregularity, we discuss how to c...

Nonadjacent Radix-τ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

In the seminal papers [6, 7], Koblitz curves were proposed for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the number fields Q( √ −3) and Q( √ −7). The (window) nonadjacent form of τ -expansion of integers in Q( √ −7) was first investigated in [11]. For integers in Q( √ −3), the nonadjacent form and the wind...