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.

       A new class of biologically active nanocomposites and modified polymers based on poly (vinyl alcohol) (PVA) with some organic compounds [II, IV, V and VI] were synthesized using silver nanoparticles (Ag-NPs). All compounds were synthesized using nucleophilic substitution interactions and characterized by FTIR, DSC and TGA. The biological activity of the modified polymers was evaluated against: gram (+) (staphylococcus aureus) and gram (-): (Es cherichia coli bacteria). Antimicrobial films are developed based on modified poly (vinyl alcohol) MPVA and Ag-NPs nanoparticles. The nanocomposites and modified polymers showed better antibacterial activities against Escherichia coli (Gram negative) than against Staphyloc

A novel Schiff base (SB) ligand, abbreviated as HDMPM, resulted from the condensation of 2-amino-4-phenyl-5-methyl thiazole and 4-(diethylamino)salicyaldehyde, and its metal complexes with [Co(II), Cu(II), Ni(II), and Zn(II)] ions in high yield were formed. The physico-chemical techniques such as elemental analysis, molar conductance, IR, 1H and 13C NMR, mass spectroscopy, and electronic absorption studies were utilized to characterize the synthesized compounds. The studied compounds were examined for their possible anticancer activity against a number of human cancerous cell lines, including A549 lung carcinoma, HepG2 liver cancer, HCT116 colorectal cancer, and MCF-7 breast cancer cell lines, with doxorubicin serving as the standard. The s

Background: Prostate cancer (PCa) is the most prevalent non-cutaneous malignancy and the second-leading cause of cancer-relatedmortality among males. Cancer is just one of the several disorders that may be linked to variations in trace element concentrations.Objective: To evaluate serum arrestin β-2 as a prostate cancer tumor marker and its association with trace elements.Methods: This case- control study was carried out at the Department of Biochemistry, College of Medicine, University of Baghdad and at the Urology department, Ghazi Al-Hariri Hospital for Surgical Specialities during the period from March 2022 to May 2023. In this case-control study 120 males were enrolled: Sixty men with newly diagnosed primary prostate cancer (P

The research includes the synthesis and identification of the mixed ligands complexes of M+2ions in general composition[M(Asn)2(SMX)] Where L- Aspargine (C4H8N2O3)symbolized (AsnH) as a primary ligand and Sulfamethoxazole(C10H11N3O3S) symbolized (SMX) as a secondary ligand. The ligands and the metal chlorides were brought in to reaction at room temperature in(v/v) ethanol /water as solvent containing NaOH. The reaction required the following [(metal: 2(Na+Asn-): (SMX)] molar ratios with M(II) ions, Where: M(II)=Mn(II), Co(II), Ni(II), Cu(II), Zn(II), Cd(II) and Hg(II). The UV–Vis and magnetic moment data revealed an octahedral geometry around M(II), The conductivity data show a non-electrolytic nature of the complexes. The antimicrobial a

Heterocyclic systems, which are essential in medicinal chemistry due to their promising cytotoxic activity, are one of the most important families of organic molecules found in nature or produced in the laboratory. As a result of coupling N-(4-nitrophenyl)-3-oxo-butanamide (3) using thiourea, indole-3-carboxaldehyde, or piperonal, the pyrimidine derivatives (5a and 5b) were produced. Furthermore, pyrimidine 9 was synthesized by reacting thiophene-2-carboxaldehyde with ethyl cyanoacetate and urea with potassium carbonate as a catalyst. The chalcones 11a and 11b were synthesized by reacting equal molar quantities of 1-naphthaldehy

The synthesized ligand [4-chloro-5-(N-(5,5-dimethyl-3-oxocyclohex-1-en-1-yl)sulfamoyl)-2-((furan-2-ylmethyl)amino)benzoic acid] (H2L1) was identified utilizing Fourier transform infrared spectroscopy (FT-IR), 1 H, 13 C – NMR, (C.H.N), Mass spectra, UVVis methods based on spectroscopy. To detect mixed ligand complexes, analytical and spectroscopic approaches such as micro-analysis, conductance, UV-Visible, magnetic susceptibility, and FT-IR spectra were utilized. Its mixed ligand complexes [M(L1)(Q)Cl2] [ where M= Co(II), Ni(II) , and Cd(II)] and complexes [Pd(L1)(Q)] and [Pt(L1)(Q)Cl2]; [H2L1] =β-enaminone ligand =L1 and Q= 8-Hydroxyquinoline = L2]. The results showed that the complexes were synthesised utilizing the molar ratio M: L1

In this work, a series of new maleimides linked to substituted benzothiazole moiety were synthesized. Synthesis of these new cyclic imides were performed via three steps, the first one involved preparation of a series of 2-aminobenzothiazole substituted with different substituents via reaction of different primary aromatic amines with ammonium thiocyanate and bromine in glacial acetic acid. The prepared 2- amino benzothiozoles were introduced in the second step in reaction with maleic anhydride producing a series of N-(substituted benzothiazole-2-yl) maleamic acids.The resulted maleamic acids were dehydrated in the third step via treatment with acetic anhydride and anhydrous sodium acetate to afford a series of the desirable N-(substitu