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.
Collective decisions to retrofit multi-family residential buildings require co-owners to agree on how the total cost is divided among dwellings, yet the distributional properties of alternative allocation rules have been insufficiently investigated at scale. Using harmonised Energy Performance Certificate (EPC) microdata covering over 4 million apartments in almost 450,000 buildings across Poland, England and Wales, and the Netherlands, we simulate five allocation rules: area-proportional, progressive-area, emissions-proportional, inefficiency-proportional, and Shapley-value allocation. For each building, we evaluate the resulting cost-share distributions using within-building inequality indices, size-progressivity measures, and cooperative-game-theoretic stability criteria. We find that performance-based rules produce within-building Gini coefficients 2 to 11 times higher than area-proportional allocation, with systematic variation across national building stocks. These rules are also less proportional in terms of dwelling size, assigning larger cost shares to smaller dwellings than their floor-area shares warrant. The Shapley rule, often advocated on cooperative-game-theoretic grounds, routinely allocates more to a single dwelling than that dwelling's stand-alone retrofit cost. The results show that cost-allocation rules generate distributional and stability consequences at policy-relevant magnitudes. For retrofit governance in multi-owner buildings, allocation design should therefore be treated as a central component of policy implementation rather than a technical-administrative choice.
Recent research documents the growing frequency of land-use conflicts in rural areas. As conflicts between farmers and other stakeholders intensify while farmland resources continue to decline, spatial planning and the legal protection of agricultural land have become key policy priorities. This makes understanding farmers’ preferences regarding neighboring land uses particularly important. However, quantitative evidence remains limited. To address this gap, we used a discrete choice experiment to examine farmers’ perceptions of potentially conflict-generating neighboring land uses in a sample of 960 farmers from the Warmińsko-Mazurskie voivodeship, Poland. The results show that residential and business activities, especially when conducted at a large scale, are perceived as the most conflict-generating types of neighboring land uses. However, farmers’ perceptions vary across farm characteristics and locations. Residential neighborhoods are perceived as particularly conflict-generating by livestock farmers and farmers in highly urbanized municipalities, while patterns by farm size and nature-protection status are less consistent.
Excessive use of mineral fertilizers contributes to soil degradation, nutrient losses, and greenhouse gas emissions. Biological strategies that mobilize native soil nutrient pools may help reduce fertilizer inputs without compromising crop productivity. This study evaluated a defined, cell-free biofertilizer composed of siderophores and siderophore-associated metabolites (SSAM) of microbial origin as a tool to enhance soil nutrient bioavailability and reduce phosphorus (P) and potassium (K) fertilization in radish cultivation. A greenhouse experiment was conducted using Raphanus sativus L. grown under optimal and 50
Tidal disruption events (TDEs) have been proposed as valuable laboratories for studying dormant black holes. However, progress in this field has been hampered by the limited number of observed events. In this work, we present TDECat, a comprehensive catalogue of 134 confirmed TDEs (131 optical TDEs and three jetted TDEs) discovered up to the end of 2024, accompanied by multi-wavelength photometry (X-ray, UV, optical, and infrared) and publicly available spectra. We also study the statistical properties, spectral classifications, and multi-band variability of these events. Using a Bayesian Blocks algorithm, we determined the duration, rise time (trise), decay time (tdecay), and their ratio for 103 flares in our sample. We find that these timescales follow a log-normal distribution. Furthermore, our spectral analysis shows that most optical TDEs belong to the TDE-H+He class, followed by the TDE-H, TDE-He, and TDE-featureless classes, which is consistent with expectations from main-sequence star disruption. Using archival observations, we identified three new potentially repeating TDEs, namely, AT 2024pvu, AT 2022exr, and AT 2021uvz, increasing the number of known repeating events. In both newly identified and previously known cases, the secondary flares exhibit a similar shape to the primary. We also examined the infrared and X-ray emission from the TDEs in our catalogue, and find that 14 out of the 18 infrared events have associated X-ray emission, strongly suggesting a potential correlation. Finally, we find that for three sub-samples (repeating flares, infrared-emitting events, and X-ray-emitting events), the spectral classes are unlikely to be randomly distributed, suggesting a connection between spectral characteristics and multi-wavelength emission. TDEcat enables large-scale population studies across wavelengths and spectral classes, providing essential tools for navigating the data-rich era of upcoming surveys such as the Legacy Survey of Space and Time.
