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.

Coupling reaction of 2-amino benzoic acid with phenol gave the new bidentate azo ligand. The prepared ligand was identified by Microelemental Analysis, FT-IR and UV-Vis spectroscopic technique. Treatment of the prepared ligand with the following metal ions (CoII, NiII, CuII and ZnII) in aqueous ethanol with a 1:2 M:L ratio and at optimum pH, yielded a series of neutral complexes of the general formula [M(L)2]. The prepared complexes were characterized using flame atomic absorption, (C.H.N) Analysis, FT-IR and UV-Vis spectroscopic methods 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 concentr

Complexes of Co(II),Ni(II),Cu(II) and Zn(II) with mixed ligands of phenylalanine (L) and tributylphosphine (TBPh) were prepared in aqueous ethanol with (2:1:1) (M:L:TBPh). The prepared complexes were characterized using flame atomic absorption,(C.H.N)Analysis, FT.IR and UV-Vis spectroscopic methods as well as magnetic susceptibility and conductivity measurements. In addition biological activity of the phenylalanine and complexes against two selected type of bacteria were also examined. Some of the complexes exhibit good bacterial activities. From the obtained data the octahedral structure was suggested for all prepared complexes.

Coupling reaction of ( 4-amino antipyrene) with the (L- tyrosine ) gave the new azo ligand 2- ( 4- Antipyrene azo ) - tyrosine .Treatment of this ligand with metal ions (Mn(II) ,Co(II), Ni(II), and Cu(II) )in ethanolic medium in (1:2) (M:L) ratio yield a series of a neutral complexes of the general formula [M(L)2] . The prepared complexes were characterized using flame atomic absorption , FT.IR , UV-Vis spectroscopic and elemental microanalysis (C.H.N) as well as magnetic susceptibility and conductivity measurement

A new mixed ligand complexes have been prepared between 8- hydroxy quinoline and o-hydroxybenzylidene-1-phenyl-2,3-dimethyl-4-amino-3-pyrazolin-5-on with Mn(II),Fe(II),Co(II),Ni(II) and Cu(II) ions . the prepared complexes were isolated and characterized by (FT-IR)and (UV-Vis) spectroscopy. Elemental analysis (C.H.N) Flame atomic absorption technique . in addition to magnetic susceptibility and conductivity measurement.

Newly prepared derivatives of Heterocyclic of dicarboxylic acid include 1, 2, 4-Triazoledicarboxylic acid. Thiocarbohydrazine (TCH) reacts with aliphatic and aromatic dicarboxylic acids, and when these resulting compounds interact with compounds containing a group of carbonyl they result in Schiff base, which are very important in the industrial and medical fields and the acids used (oxalic acid, succinic, terephthalic) to prepare the triazole, then the reaction with Para-chlorobenzendihaide. and some physical properties were measured for these products. The biological activity of the prepared compounds has been studied, and it has been shown that they have different effects on the bacteria, compounds prepared with Fourier Transform Infrare

The free Schiff base ligand (HL1) is prepared by being mixed with the co-ligand 1, 10-phenanthroline (L2). The product then is reacted with metal ions: (Cr+3, Fe+3, Co+2, Ni+2, Cu+2 and Cd+2) to get new metal ion complexes. The ligand is prepared and its metal ion complexes are characterized by physic-chemical spectroscopic techniques such as: FT-IR, UV-Vis, spectra, mass spectrometer, molar conductivity, magnetic moment, metal content, chloride content and microanalysis (C.H.N) techniques. The results show the formation of the free Schiff base ligand (HL1). The fragments of the prepared free Schiff base ligand are identified by the mass spectrometer technique. All the analysis of ligand and its metal complexes are in good agreement with th

Mefenamic acid (MA) is one of the non-steroidal anti-inflammatory drugs, it is widely used probably due to having both anti-inflammatory and analgesic activity, the main side effects of mefenamic acid include gastrointestinal tract (GIT) disturbance mainly diarrhea, peptic ulceration, and gastric bleeding. The analgesic effects of NSAIDs are probably linked to COX-2 inhibition, while COX-1 inhibition is the major cause of this classic adverse effects. Introduction of thiazolidinone may lead to the increase in the bulkiness leads to the preferential inhibition of COX-2 rather than COX-1 enzyme. The study aimed to synthesize derivatives of mefenamic acid with more potency and to decrease the drug's potential side effects, new series of 4-t

6-(2-benzathiazolyl azo),-3,5-dimethylphenol was formed by grouping the 2- benzothiazole diazonium chloride with 3,5-dimethylphenol. Azo ligand(L) was resolved on the origin by 1H and 13CNMR, FTIR and UV-V is spectral analysis. Complexation of tridentate ligand (L) with Co2+, Ni2+, Cu2+ and Zn2+ in aqueous of ethyl alcohol with a 1:2 metal:ligand, and at ideal pH.. The formation of metal chelates are assigned using flame atomic, absorption, FTIR, and UV-Vis spectral analysis, other than conductivity and magnetic estates. The nature of the metal chelates were carried out by mole ratio and continuous, variation mechanism, Beer's law, followed the rate (0.0001 - 3×0.0001 M) concentration., High molar, absorptivity, for the complex solutions w