Gaussian Integer Solutions of the Diophantine Equation x^4+y^4=z^3 for x≠ y

Shahrina Ismail; Kamel Ariffin Mohd Atan; Diego Sejas Viscarra; Kai Siong Yow

doi:10.21123/bsj.2023.7344

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Publication Date

Sun Oct 01 2023

Journal Name

Baghdad Science Journal

Volume

20

Issue Number

5

DOI

10.21123/bsj.2023.7344

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Gaussian Integer Solutions of the Diophantine Equation x^4+y^4=z^3 for x≠ y

Algebraic properties

Diophantine equation

Gaussian integer

quartic equation

nontrivial solutions

symmetrical solutions.

Shahrina Ismail

Kamel Ariffin Mohd Atan

Diego Sejas Viscarra

Kai Siong Yow

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The investigation of determining solutions for the Diophantine equation over the Gaussian integer ring for the specific case of is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

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Publication Date

Thu Apr 01 2021

Journal Name

Pakistan Journal Of Statistics

Estimation intensity radiation of chest X-ray (CXR) with application

Chest x-ray (CXR)

emissions

intensity of radiation

Sievert

Bayes

Asmaa Najm

Iqbal Mahmood

Suhad Ahmed

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In this research we assumed that the number of emissions by time (𝑡) of radiation particles is distributed poisson distribution with parameter (𝑡), where  < 0 is the intensity of radiation. We conclude that the time of the first emission is distributed exponentially with parameter 𝜃, while the time of the k-th emission (𝑘 = 2,3,4, … . . ) is gamma distributed with parameters (𝑘, 𝜃), we used a real data to show that the Bayes estimator 𝜃 ∗ for 𝜃 is more efficient than 𝜃̂, the maximum likelihood estimator for 𝜃 by using the derived variances of both estimators as a statistical indicator for efficiency

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Publication Date

Tue Feb 12 2019

Journal Name

Iraqi Journal Of Physics

A study of the structural and electrical properties of Ni1-x Cox Fe2O4 ferrites

Nickel-Cobalt Ferrite

structure

Electrical properties.

Hind Mohammed

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Ferrite with general formula Ni1-x Cox Fe2O4(where x=0.0.1,0.3,0.5,0.7, and 0.9), were prepared by standard ceramic technique. The main cubic spinel structure phase for all samples was confirmed by x-ray diffraction patterns. The lattice parameter results were (8.256-8.299 ^°A). Generally, x -ray density increased with the addition of Cobalt and showed value between (5.452-5.538gm/cm³). Atomic Force Microscopy (AFM) showed that the average grain size and surface roughness was decreasing with the increasing cobalt concentration. Scanning Electron Microscopy images show that grains had an irregular distribution and irregular shape. The A.C conductivity was found to increase with the frequency and the addition of Cobal

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Publication Date

Wed Jun 10 2015

Journal Name

Journal Of The College Of Basic Education

Synthesis, characterization and antimicrobial studies of 2-(4-antipyrine azo)-4-nitroaniline with some metal ions.

spectral studies

complexes

antimicrobial activity

Amer J.Jarad

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Transition metal complexes of Co(II), Ni(II), Cu(II), and Zn(II) with 2-(4-antipyrine azo)-4-nitroaniline derived from 4-aminoantipyrine and 4-nitroaniline were synthesized. Characterization of these compounds has been done on the basis of elemental analysis, electronic data, FT-IR, UV-Vis and 1HNMR, as well as magnetic susceptibility and conductivity measurements. The nature of the complexes formed were studied following the mole ratio and continuous variation methods, Beer's law obeyed over a concentration range (1×10-4 - 3×10-4 M). High molar absorbtivity of the complex solutions were observed. From the analytical data, the stoichiometry of the complexes has been found to be 1:2 (metal:ligand). On the basis of physicochemical data octa

Publication Date

Sun Jan 01 2017

Journal Name

Journal Of The College Of Languages (jcl)

Renaissance des Briefromans Oder Der Alltag in Krisen-und Wendezeiten

Mohammad Ismail

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In der Nachwendezeit erschienen im deutschen Buchmarkt zahlreiche Werke deutscher Autoren, die einen Trend oder Tendenz markieren können. Im Mittelpunkt der vorliegenden Arbeit steht die Analyse der Struktur und des Inhalts von drei erst nach der Wende erschienen Briefromanen: Alles, alles Liebe (2000) von Barbara Honigmann (geb. 1949), Die Liebenden (2002) von Gerhard Henschel (Jahrgang 1962) und Neue Leben (2005) des im Jahre 1962 geborenen Ingo Schulze. Dieser Beitrag macht sich zur Aufgabe zu erklären, ob der Briefroman ein neuer Trend in der deutschen Literatur nach der Wende ist.

Abstract

In the after( Wende) period a large collection of literary works for German writers has been p

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Publication Date

Wed Mar 10 2021

Journal Name

Baghdad Science Journal

studies of human interferon a B and Y activities on diffrent cell

Abdullatif Muhammad

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Human interferon as is the case in all kinds of interferon has complex effects but all share their impact on preventing the proliferation of viruses and preventing or reducing human Alantervjørn conversion occurs if the cell is in preventing the growth of the virus when interferon Balnmstqubl connects

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Publication Date

Sat Sep 01 2018

Journal Name

Polyhedron

Novel dichloro (bis {2-[1-(4-methylphenyl)-1H-1, 2, 3-triazol-4-yl-κN3] pyridine-κN}) metal (II) coordination compounds of seven transition metals (Mn, Fe, Co, Ni, Cu, Zn and Cd)

The 1

3-dipolar cycloaddition “click” reaction between an azide and an alkyne to give a 1

2

3-triazole was reported by Huisgen in 1961 [1]. In 2002 the Copper(I)-catalyzed Azide-Alkyne Cycloaddition (CuAAC) to prepare a 1

2

3-triazole was reported [2]

[3]. The existence of relatively basic nitrogen atoms in the 1

2

3-triazole rings

and the possibility of introducing additional donor groups in the substituents (Fig. 1)

made the CuAAC “click” reaction an attractive method to prepare differently substituted 1

2

3-triazoles. These compounds have been used as ligands to coordinate to various metal ions that display a range of applications such as in electrochemical and photochemical studies

in supramolecular chemistry

magnetism

metal-ion sensing and catalysis [4]. The reasons for the success of the “click” reaction

is that it is easy to carry out and is widely applicable. It is not affected by a variety of functional groups

and can be carried out with a variety of Cu(I) catalysts and solvents

including aqueous conditions. The Cu(I) catalysts overcome the high activation energy barrier of the non-catalyzed Huisgen reaction by changing the mechanism of the reaction. A large variety of copper catalysts can be used for the CuAAC reaction

on condition that the maximum concentration of Cu(I) species is generated during the reaction. The pre-catalyst can be a Cu(II) salt (usually CuSO4) together with a reducing agent (often sodium ascorbate) or a Cu(I) compound in the presence of a base or amine ligand and a reducing agent to prevent oxidation to Cu(II). Some strong oxidising cupric salts or complexes such as Cu(OAc)2 also work. The solvent is very flexible from organic to aqueous

with the most commonly used combination water + an alcohol (t-BuOH

MeOH or EtOH). The key role of the solvent or solvent mixture is to solubilize the substrates and Cu(I) catalyst in order to ensure rapid reactions. Such aqueous conditions are very useful for biochemical conjugations

as well as for organic syntheses. We recently reported on the synthesis of a series of differently substituted 1

2

3-triazole chromophores

the substituted 2-(1-phenyl-1H-1

2

3-triazol-4-yl)pyridine ligands [5]

see Fig. 2 middle

with substituents R = H (L1)

CH3 (L2)

OCH3 (L3)

COOH

F

Cl

CN

CF3

O(CH2)3CH3 and N(CH3)2. These versatile ligands were found to coordinate to various first row transition metals

such as manganese

cobalt and nickel [6]. Here we extend the series to include more first row transition metal(II) coordination compounds

iron

copper and zinc

as well as a second row transition metal(II) coordination compound

cadmium

containing the 2-(1-(4-methyl-phenyl)-1H-1

2

3-triazol-1-yl)pyridine chromophore (Fig. 2 right with R = CH3). This series of seven novel coordination compounds is the first series of pyridyl-triazole based transition metal coordination compounds where seven different transition metals are coordinated to the same 1

2

3-triazole chromophore

namely 2-(1-(4-methyl-phenyl)-1H-1

2

3-triazol-1-yl)pyridine.

Kinaan M

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Publication Date

Sun Dec 18 2022

Journal Name

Journal Of The College Of Basic Education

A (k,ℓ) Span in Three Dimensional Projective Space PG(3,p) Over Galois Field where p=4

Fatema Faisal

Sawsan

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الغرض من هذا العمل هو دراسة الفضاء الإسقاطي ثلاثي الأبعاد PG (3، P) حيث p = 4 باستخدام المعادلات الجبرية وجدنا النقاط والخطوط والمستويات وفي هذا الفضاء نبني (k، ℓ) -span وهي مجموعة من خطوط k لا يتقاطع اثنان منها. نثبت أن الحد الأقصى للكمال (k، ℓ) -span في PG (3،4) هو (17، ℓ) -span ، وهو ما يساوي جميع نقاط المساحة التي تسمى السبريد.

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(1)

Publication Date

Thu Oct 01 2020

Journal Name

Biochem. Cell. Arch

NEW ORGANOTIN(IV) COMPLEXES DERIVED FROM 3,4-DIHYDROXYBENZALDEHYDEN(4)-ETHYL-3-SEMICARBAZONE LIGAND: SYNTHESIS, CHARACTERISATION AND BIOLOGICAL ACTIVITY

Baidaa

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Two new organotin(IV) complexes Me2Snesc (C1) and Bu2Snesc (C2) have been synthesised from the reaction of the corresponding organotin(IV) chloride with the Schiff base ligand 3,4-dihydroxybenzaldehyde-4-ethylsemicarbazone (H2esc). The ligand was prepared in two steps. The first step includes the formation of 4-ethylsemicarbazide, which then reacted with 3,4-dihydroxybenzaldehyde to give the title ligand. Complex formation between the organotin(IV) moiety and the anionic form of 3,4-dihydroxybenzaldehy-4-ethylsemicarbazone occurred through the o-dihydroxy positions. The ligand and its complexes were characterised by elemental analysis, FT-IR and NMR (1H, 13C and 119Sn) spectroscopy. Accordingly, the complexes were proposed to have tetrahedr

(6)

Publication Date

Wed May 17 2023

Journal Name

Journal Of The Australian Ceramic Society

Dielectric and scattering properties of transparent SiC composite in X-band range for radome application

Alaa Raad

...Show More Authors

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(1)

Publication Date

Wed Apr 12 2017

Journal Name

Ibn Al-haitham Journal For Pure And Applied Science

Synthesis, Characterization and Study Biological Activity of Some New 1, 3, 4-Thiadiazole and Pyrazolone Derivatives Containing Indole Ring

Khalid

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This work involves synthesis and characterization of some new 1, 3, 4-thiadiazole or pyrazoline derivatives heterocyclic containing indole ring. The new 2-amino-1, 3, 4-thiadiazole derivatives [IV] and [V] a, b were synthesized by cyclization reaction of 2-methyl-1H-indole-carbothiosemicarbazide [III] in H2SO4 acid or by reaction of indole-3-acetic acid or indole-3-butanoic acid with thiosemicarbazide in the presence of phosphorous oxychloride, respectively. Amide derivatives [VI]-[VIII] were synthesized by the reaction equimolar of 2-amino-1, 3, 4-thiadiazoles and (acetyl chloride, benzoyl chloride, anisoyl chloride and heptanoyl chloride) in DMF and pyridine as accepter. The new pyrazolone derivatives [XI] a, b were synthesized from heati

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