ON THE 82-TH SMARANDACHE’S PROBLEM

The main purpose of this paper is using the elementary method to study the asymptotic properties of the integer part of the k-th root positive integer, and give two interesting asymptotic formulae

An open problem on Jeśmanowicz\u27 conjecture concerning primitive Pythagorean triples

Let (m>31) be an even integer with (gcd(m,31)=1). In this paper, using some elementary methods, we prove that the equation ((m^2-31^2)^x+(62m)^y=(m^2+31^2)^z) has only the positive integer solution ((x,y,z)=(2,2,2)). This result resolves an open problem raised by T. Miyazaki ({em Acta Arith.} 186 (2018), 1--36) about Je\u27smanowicz\u27 conjecture concerning primitive Pythagorean triples

Vitamin C Promotes Wound Healing: The Use of in Vitro Scratch Assays to Assess Re-Epithelialization

Vitamin C contributes to the health of the cardiovascular, immunological and nervous system and also supports healthy bone, lung, and skin function and wound healing. The mechanism of action of vitamin C in human health is as diverse as its targets and effects. For example, vitamin C functions as an antioxidant, signals changes in gene expression, plays a role in protein metabolism, and serves a cofactor in several biosynthetic pathways including collagen synthesis. Here in this chapter we will explore the role of vitamin C in support of improved wound healing during the re-epithelialization stage. While vitamin C supports wound healing in early stages by reducing inflammation, vitamin C continues to support wound healing by promoting collagen synthesis and epithelial cell migration in the re-epithelialization stage. The re-epithelialization stage of wound healing has been modeled and investigated using an in vitro scratch-assay in which a monolayer of epithelial cells is scratched to create a gap or void in the monolayer to represent the wound. The rate of epithelial cell migration back across this gap to re-establish the monolayer can then be used as a model and measurement of the re-epithelialization stage of wound healing. Again, this Chapter will review the literature on both a) the uses of in vitro scratch assays to investigate the mechanism of vitamin C enhanced epithelial cell migration and b) the potential uses of the in vitro scratch assay to study the bioavailability and absorption of liposomal vitamin C

Integral points on the elliptic curve y2=x3−4p2xy^2=x^3-4p^2x

summary:Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E\colon y^2=x^3-4p^2x$. Further, let $N(p)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\geq 17$, then $N(p)\leq 4$ or $1$ depending on whether $p\equiv 1\pmod 8$ or $p\equiv -1\pmod 8$

ON THE 82-TH SMARANDACHE’S PROBLEM

Keywords: The main purpose of this paper is using the elementary method to study the asymptotic properties of the integer part of the k-th root positive integer, and give two interesting asymptotic formulae. k-th root; Integer part; Asymptotic formula

Perfect codes in power graphs of finite groups

The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes