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.
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.
PurposeThe aim of this paper was to investigate the impact of blockchain technology on firm value, using the mediating role of corporate governance.Design/methodology/approachTo test the hypotheses, the authors applied linear regression with panel data using the Thomson Reuters ASSET4 on a sample of 197 companies selected from the environmental, social and governance index over the period 2010-2023. To estimate the econometric models, the feasible generalised least squares regression method was employed. To enhance the robustness of the results, alternative measures of the mediating variable and sample heterogeneity are employed. Furthermore, a structural equation model was employed for additional analysis.FindingsThe empirical results indicate a positive impact of blockchain technology on firm value. Furthermore, the results suggest that corporate governance positively mediates the relationship between blockchain technology and firm value.Originality/valueThis paper investigates this relationship through the mediating role of corporate governance, where previous studies have explored the direct relationship between blockchain technology and firm value. The study highlights the importance of corporate governance and technological advancement and argues that regulators and service providers should promote the integration of blockchain technology to enhance firm value.
