Let Ps(n) denote the n-th s-gonal number. Consider the Diophantine equation Ps(n)=tm for integers n,s,t and m>2. All solutions to this equation are known for m>2 and s∈{3,5,6,8,10,20}. Here we extend these results to the cases s=2k+4 (where k=4,6 or 5≤k≤97 is a prime number) and s=k+4 (where k=9,15 or 3≤k≤97 is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective abc conjecture) that there will be no additional solutions beyond those explicitly shown in Theorem 1, Theorem 2, Theorem 3.
Background: Chatbot become more popular, marketing research has paid enormous attention to chatbot and its impact on customer’s intention.Aims: This article aims to reveal the utilitarian motivations and communication style of the chatbot by highlighting the role of social presence and its impacts on user intent.Methodology: In this study, by applying social presence theory, adding utilitarian motivation and communication style as explicative variables, a survey data of 200 users of chatbot was used. The theoretical model was tested and confirmed via structural equation modelling (SEM).Findings: The results indicated that the utilitarian motivations, communication style and social presence of the chatbot are the main determinants of the intention to use this electronic channel. Theoretical and managerial implications as well as the limits and future directions of the study are discussed.Conclusion: This study examines the impact of chatbots on consumer intent, highlighting the role of social presence and communication style. The results confirm that these factors positively influence user intentions.
This study explores how the green backgrounds of female executives (GFEMs) influence corporate climate targets disclosure (CTD). Using a panel dataset comprising China's A-share nonfinancial listed companies (2010-2022) and rigorous methodologies, the findings confirm that GFEMs promote CTD. We find a nonlinear correlation with CTD rising with GFEM representation to an optimal point, but again weakening due to over-concentration of GFEM presence, an inverted U-shaped association reminiscent of critical mass dynamics. Moreover, mediation analysis revealed that GFEMs influence corporate CTD by adopting a low-carbon strategy and management green attention, influencing corporate strategic and cognitive responses to climate change commitments. The additional results indicate that the impact of GFEMs on CTD depends on the gender alignment of the executive teams and transpires under the leadership of male CEOs. This study offers comprehensive insights into the interrelationships among individual executive traits, leadership structures, and climate accountability.
This study examines the structural, optical, and electrical properties of NaBa(PO3)3 to evaluate its potential in solid-state ionic devices. Phase-pure NaBa(PO3)3 was synthesized by a high-temperature solid-state reaction and characterized using XRD, FTIR, UV–Vis, and AC impedance spectroscopy. Rietveld refinement confirmed the formation of an orthorhombic P212121 structure. The material exhibits a wide optical band gap and thermally activated Na+ conduction consistent with polaron hopping, highlighting its potential for high-temperature ionic and dielectric applications. Impedance and modulus analyses reveal non-Debye relaxation and distinct grain and grain-boundary contributions. Its negative temperature coefficient of resistivity and high sensitivity index highlight strong potential for high-temperature thermistor and solid-state electrolyte applications.
Bacterial contamination is a persistent issue in food preservation, medical hygiene, and water purification, necessitating the development of eco-friendly and effective antibacterial solutions. In this context, silver nanoparticles (AgNPs) were biosynthesized using an aqueous extract of Marrubium vulgare L. leaf powder - a plant scarcely explored for nanoparticle synthesis. The process was optimized to enhance surface plasmon resonance (SPR) using a Box-Behnken design based on response surface methodology (RSM), taking into account silver nitrate concentration, extract-to-silver ratio, temperature, and reaction time. The statistical model showed impressive prediction capacity (R-2 > 98 %), which was experimentally validated, with high reproducibility. The biosynthesized AgNPs exhibited a well-defined SPR peak around 440 nm, appearing as spherical, uniformly dispersed particles with an average size of 35.35 +/- 12.56 nm. XRD analysis confirmed their crystalline structure, while a zeta potential of -48.5 mV indicated excellent colloidal stability - a finding reinforced through assessments over time, demonstrating that the nanoparticles remain highly stable and easily reproducible. The involvement of plant-derived biomolecules was thoroughly examined, revealing that significant proportions of polyphenols (82.07 %), flavonoids (88 %), and tannins (45.49 %) contributed to nanoparticle reduction and stabilization. The antibacterial activity of the synthesized nanoparticles was tested against both Gram-positive and Gram-negative bacteria, showing effective inhibition with zones ranging from 7.23 +/- 0.59 mm to 10.60 +/- 0.81 mm. Their consistent performance across different bacterial types highlights their versatility and strong potential as a multifunctional antibacterial solution. Overall, these findings position M. vulgare-based AgNPs as a highly stable, reproducible, and eco-friendly alternative for alleviating bacterial contamination in various applications.
