Quadratic non-residues and non-primitive roots satisfying a coprimality condition

Let $q\geq 1$ be any integer and let $\epsilon \in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-\epsilon} \mbox{ and } \frac{\phi(p-1)}{p-1} \leq \frac{1}{2} - \epsilon, there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$.Comment: to appear in Bulletin of the Australian Mathematical Societ

MANAGEMENT OF MICROBIAL BIOFILM USING NANO PARTICLE: A REVIEW

Microorganisms create biofilms, which are surface adherent community structures. These biofilms are essential to the infection process mediated by microbes. Antibiotic resistance is another thing that biofilm spreads, which is a big worry these days. Diverse bacteria use diverse mechanisms to create biofilms, and these mechanisms often depend on the environment in which they grow as well as strain-specific characteristics. Many chemical compounds are discovered to be useful in investigating the biofilm management method. The usefulness of nanoparticles in preventing biofilm-mediated disease is the subject of the current review. Using nanoscale particles to fight microbial biofilm is one possible way to treat these persistent diseases. Recently, antibacterial agents have been delivered employing innovative nanotechnology-based antimicrobial activity in order to destroy planktonic bacteria and their biofilm structures. In the sphere of medicine, this technique is now considered developing. Antimicrobial-loaded nanoparticles alone or in combination with other materials could increase the bacterial activity of nanomaterials to prevent the formation of biofilms. These particles are reactive substances that readily penetrate the matrix, serving as a barrier to numerous antibodies. One type of nanoparticle, called AgNPs, exhibited antibacterial action by rupturing the integrity of the bacterial cell membrane, which resulted in the release of cellular content and eventual death. Additionally, polymeric-based formulations like hydrogel, polymeric microspheres, nanospheres, and smart olimer, as well as lipid-based nanoparticles like liposomes and solid lipid nanoparticles, have been used in the biofilm treatment. Additionally, research is ongoing with various metals like copper, zinc, and their oxides. Here, we talked about the safety issues and the promise of metal oxide nanoparticles. The pathogens are effectively killed by NPs without endangering other cells or having any negative effects on living cells