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.

Tin:antimony oxides composites have given great attention due to their unique properties and many application in various fields like optoelectronic devices, gas sensors. The synthesis and characterization of (SnO2)1-x(Sb2O3)x was conducted in order to comprehend the structure and discuss the optical and dielectric properties of this composites. The x-ray showed that the prepared pure tin oxide thin film was had a polycrystalline structure and that the peaks were identical with the tetragonal phase while the peaks were identical with antimony oxide for x=0.1 to x=0.5. The average crystal size along the preferred level of growth of tin oxide as well as of antimony oxide showed non regular change by i

We report about multilayer (TiO2)1-x(WO3)x thin films with various tungsten oxide concentration deposited on glass substrate via pulsed laser deposition (PLD) technique method. The structural, optical and dielectric properties of the prepared thin films were characterized using x-ray diffraction method (XRD) and UV-Visible absorption spectrometry and LRC meter. The x-ray results showed a polycrystalline structure of pristine titanium oxide and the peaks are identical with the rutile phase and trace of the anatase phase. These peaks are identical with titanium oxide and triclinic tungsten oxide as WO3 is added to TiO2 while it was observed that all peaks belong to tungsten oxide at x=1.0. Th

Silver selenide telluride Semiconducting (Ag2Se0.8Te0.2) thin films were by thermal evaporation at RT with thickness350 nm at annealing temperatures (300, 348, 398, and 448) °K for 1 hour on glass substrates .using X-ray diffraction, the structural characteristics were calculated as a function of annealing temperatures with no preferential orientation along any plane. Atomic force microscopy (AFM) and X-ray techniques are used to analyze the Ag2SeTe thin films' physical makeup and properties. AFM techniques were used to analyze the surface morphology of the Ag2SeTe films, and the results showed that the values for average diameter, surface roughness, and grain size mutation increased with annealing temperature (116.36-171.02) nm The transm

Silver selenide telluride Semiconducting (Ag2Se0.8Te0.2) thin films were by thermal evaporation at RT with thickness350 nm at annealing temperatures (300, 348, 398, and 448) °K for 1 hour on glass substrates .using X-ray diffraction, the structural characteristics were calculated as a function of annealing temperatures with no preferential orientation along any plane. Atomic force microscopy (AFM) and X-ray techniques are used to analyze the Ag2SeTe thin films' physical makeup and properties. AFM techniques were used to analyze the surface morphology of the Ag2SeTe films, and the results showed that the values for average diameter, surface roughness, and grain size mutation increased with annealing temperature (116.36-171.02) nm The transm

Thin films of ZnSxSe1-x with different sulfide content(x)
(0, 0.02, 0.04, 0.06, 0.8, and 0.1), thickness (t) (0.3, 0.5, and 0.7 μm) and annealing temperature (Ta) (R.T 373 and 423K) were fabricated by thermal evaporating under vacuum of 10-5 Toor on glass substrate. The results show that the increasing of sulfide content (x)and annealing temperature lead to decrease the d.c conductivity σDC of and concentration of charge carriers (nH) but increases the activation energy (Ea1,Ea2), while the increasing of t increases σDC and nH but decrease (Ea1,Ea2). The results were explained in different terms

In this study, thin films of pure titanium dioxide (TiO2) and titanium dioxide dual mixed with zinc oxide (ZnO) and magnesium oxide (MgO) with varying concentrations of (ZnO: MgO)x ranging from 0 to 30 wt% undoped and doped gold nanoparticles (AuNPs) were prepared on glass using the chemical spray pyrolysis (CSP) technique. The morphological, structural, and sensing properties of the prepared thin films were examined. Atomic Force Microscopy (AFM) analysis revealed that these films exhibit a consistent structure before and after doping. Initially, the roughness of these films was observed to increase upon the introduction of impurities (ZnO: MgO)x. This trend reversed at x=0.20, where a decrease in roughness occurred. Interestingly, a sub

In this study, thin films of pure titanium dioxide (TiO 2 ) and titanium dioxide dual mixed with zinc oxide (ZnO) and magnesium oxide (MgO) with varying concentrations of (ZnO: MgO) x ranging from 0 to 30 wt% undoped and doped gold nanoparticles (AuNPs) were prepared on glass using the chemical spray pyrolysis (CSP) technique. The morphological, structural, and sensing properties of the prepared thin films were examined. Atomic Force Microscopy (AFM) analysis revealed that these films exhibit a consistent structure before and after doping. Initially, the roughness of these films was observed to increase upon the introduction of impurities (ZnO: MgO) x . This trend reversed at x = 0.20, where a decrease in roughness occurred. Interestin

Thin films of tin sulfide (SnS) were prepared by thermal evaporation technique on glass substrates, with thickness in the range of 100, 200 and 300nm and their physical properties were studied with appropriate techniques. The phase of the synthesized thin films was confirmed by X-ray diffraction analysis. Further, the crystallite size was calculated by Scherer formula and found to increase from 58 to 79 nm with increase of thickness. The obtained results were discussed in view of testing the suitability of SnS film as an absorber for the fabrication of low-cost and non toxic solar cell. For thickness, t=300nm, the films showed orthorhombic OR phase with a strong (111) preferred orientation. The films deposited with thickness < 200nm deviate

Alloys were performed from In2O3 doped SnO2 with different doping ratio by quenching from the melt technique. Pulsed Laser Deposition PLD was used to deposit thin films of different doping ratio In2O3 : SnO2 ( 0, 1, 3, 5, 7 and 9 % wt.) on glass substrate at ambient temperature under vacuum of 10 -3 bar thickness of ~100nm. The structural type ,grain size and morphology of the prepared alloys compounds and thin films were examined using X-ray diffraction and atomic force microscopy . The results showed that all alloys have polycrystalline structures and the peaks belonged to the preferred plane for crystal growth were identical with the ITO (Indium – Tin –Oxide) standard cards also another peaks were ob