The aim of the present article is to give a complete list of the integer solutions to polynomial-exponential Diophantine equations, as well as those to exponential Diophantine equations in three variables. Since there is no universal effective method to determine the integer solutions to these equations, our results give new examples in the cases where the theory of linear forms in logarithms cannot be applied.

One of layered materials (BiO) CuCh (Ch = Se, Te) has been reported to a good thermoelectric semiconductor. On the other hand, (LaO) CuCh with the same crystal structure does not show such a good property. In this study, we have investigated the origin of this difference on the thermoelectric properties between them through photoemission spectroscopy using synchrotron X-ray, which can evaluate the valence band structure. Next, we fabricated novel plasmonic nanostructures consisting of Ag and Cu to efficiently harvest solar light. We found that the wavelengths of localized surface plasmon resonance of Cu periodic structures could be tuned over wide wavelengths from visible to near-infrared (NIR) region. Next, we demonstrated that the electromagnetic fields generated via the Cu LSPR excitation were quite strong by investigating the photoelectric conversion phenomenon of dye molecules immobilized on the structures. Furthermore, it was found that the hybrids consisting of Cu₂O ultrathin layers and Cu periodic structures could be acted as superior photocatalysts by the irradiation of solar light. These results suggest that the plasmonic Cu nanostructures are very useful for the fabrication of solar light-driven devices such as hydrogen generation and so on. Finally, we developed the fabrication procedures of plasmonic Ag nanoprisms with various aspect ratios. The LSPR wavelengths of these nanoparticles can be tuned by adjusting their aspect ratio. These results suggest that the Ag nanoprisms are useful nanomaterials for the efficient absorption of solar light. Furthermore, we succeeded in enhancing the upconverted emission based on the triplet-triplet annihilation.