The connection between $P$-fractions and associated fractions

A Connection between Covers of the Integers and Unit Fractions

For integers a and n > 0, let a(n) denote the residue class {x ∈ Z : x ≡ a (mod n)}. Let A be a collection {as(ns)}s=1 of finitely many residue classes such that A covers all the integers at least m times but {as(ns)} s=1 does not. We show that if nk is a period of the covering function wA(x) = |{1 6 s 6 k : x ∈ as(ns)}| then for any r = 0, . . . , nk−1 there are at least m integers in the form...

A Connection between Covers of Z and Unit Fractions

is usually called the covering function of A. Clearly wA(x) is periodic modulo the least common multiple NA of the moduli n1, . . . , nk. As in [S99] we call m(A) = minx∈Z wA(x) the covering multiplicity of A. Let m ∈ Z. If wA(x) > m for all x ∈ Z, then we call A an m-cover of Z. If A is an m-cover of Z but At = {as(ns)}s∈[1,k]\{t} is not (where [a, b] = {x ∈ Z : a 6 x 6 b} for a, b ∈ Z), then ...

p-Adic Egyptian Fractions

P -adic Continued Fractions

Continued fractions in R have a single definition and algorithms for approximating them are well known. There also exists a well known result which states that √ m, m ∈ Q, always has a periodic continued fraction representation. In Qp, the field of p-adics, however, there are competing and non-equivalent definitions of continued fractions and no single algorithm exists which always produces a p...

Shapes of Distant Clusters : the Connection to Blue Galaxy Fractions

Based on ROSAT PSPC pointed observations, we have determined the aggregate X-ray shapes of 10 distant (z = 0. Four of the clusters have global X-ray ellipticities ∼ > 0.2, as measured on a scale of diameter ∼ 3h −1 50 Mpc. These strongly elongated clusters tend to show substantial amounts of substructure, indicating that they are dynamically young systems. Most interestingly, the global X-ray e...