We compute primes $p \equiv 5 \bmod 8$ up to $10^{11}$ for which the Pellian equation $x^2-py^2=-4$ has no solutions in odd integers; these are the members of sequence A130229 in the Online Encyclopedia of Integer Sequences. We find that the number of such primes $p\leqslant x$ is well approximated by

$$ \begin{align*}\frac{1}{12}\pi(x) - 0.037\int_2^x \frac{dt}{t^{1/6}\log t},\end{align*} $$

where $\pi (x)$ is the usual prime counting function. The second term shows a surprising bias away from membership of this sequence.

We consider interacting urns on a finite directed network, where both sampling and reinforcement processes depend on the nodes of the network. This extends previous research by incorporating node-dependent sampling and reinforcement. We classify the sampling and reinforcement schemes, as well as the networks on which the proportion of balls of either colour in each urn converges almost surely to a deterministic limit. We also investigate conditions for achieving synchronisation of the colour proportions across the urns and analyse fluctuations under specific conditions on the reinforcement scheme and network structure.

Dark patterns that manipulate consumer behaviour are now a pervasive feature of digital markets. Depending on the choice architecture utilised, they can affect the perception, behaviour and purchasing patterns of online consumers. Using a novel empirical design, we find strong evidence that individuals across all groups are susceptible to dark patterns, and only weak evidence that user susceptibility is materially affected by commonly used general proxies for consumer vulnerability (such as income, educational attainment or age). Our conclusions provide empirical support for broad restrictions on the use of dark patterns, such as those contained in the EU’s Digital Services Act, that protect all consumer groups. Our study also finds that added friction, in the form of required payment action following successful deployment of dark patterns, reduces their effectiveness. This insight highlights the instances in which dark patterns would be most effective – when no further action is required by the user. Consumer vulnerability is therefore more pronounced when dealing with online providers who store users’ payment details and can rely on a ‘single click’ to complete the purchase.

Veterinarians are custodians of animal welfare, ensuring practices remain current and effective in the face of the ever-changing demands of the profession. Continuing Professional Development (CPD) is essential for protecting animal welfare, however access to quality CPD is a challenge in many countries. India has one of the fastest growing veterinary industries globally, emphasising the importance of accessible CPD opportunities that are relevant to this setting. This study used focus groups to explore how Indian veterinarians identify relevant CPD, barriers they encounter, and their experiences with CPD. We describe three themes: (1) ‘career vs calling’, where veterinarians’ extrinsic and intrinsic motivational factors were identified, such as their desire to protect animal welfare; (2) being ‘willing to learn but can’t’, with context-specific barriers, such as accessing reliable CPD information; and (3) ‘real interactions matter’, where participants described how pedagogical design influenced their choices, e.g. being able to observe animal welfare improvements through practical teaching. We have three recommendations: firstly, to improve CPD learning opportunities informed by evidence-based methods, to meet knowledge and skills gaps such as the high demand for practically focused training; secondly, the development of a unified accreditation and quality assurance framework to assess content, relevance and delivery standards of available CPD options to veterinarians; and, lastly, improved support from employers to address current barriers and facilitate attendance. These findings contribute to the current knowledge gap of factors that influence Indian veterinarians’ experiences of attaining relevant, accessible CPD and makes suggestions to improve standards of veterinary care and, ultimately, patient welfare.

This essay is a write-up of my Professorial Inaugural Lecture, delivered at the London School of Economics on 9 December 2024. Herein, I describe how I became involved and have helped develop the field of behavioural public policy (BPP). I detail how the intellectual architecture of BPP – its journal, Annual International Conference and Association – came into existence, and allude to my hopes for how the field might develop as we go forward.

We prove structural results for measure-preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions, these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function, all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwiłł, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that a trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro averages.

Populists emerge when distrust of state institutions or dissatisfaction with democracy convince voters that claims about conspiring elites blocking the general will are valid. We propose that these dynamics change when populists are incumbents; once they command institutions, their sustained support becomes contingent upon trust in the new institutional order, and they are held accountable for making people think democracy is working well. Newly collected data on party populism and survey data from Latin America show that support for populist parties in the region is conditioned by satisfaction with democracy as well as the incumbency status of populists. Dissatisfied voters support populist opposition parties, but support for populist incumbents is higher among those satisfied with democracy and its institutions. While democratic deficits and poor governance provide openings for populists, populists are held accountable for institutional outcomes.

This article offers the first detailed study of a manuscript preserving notes from the Modern Greek course held in 1801-2 by Jean-Baptiste-Gaspard d'Ansse de Villoison in Paris. The notebook was kept by Karl Benedikt Hase, later Professor of Modern Greek at the French École de langues orientales vivantes, during his attendance of Villoison's course as a student. The article sketches first the historical context of the notebook, before an analysis of its contents, and finally a comparison with Hase's later published work on the primary text at its core, Amiras’ Greek translation of Costin's History of Moldavia.

In rural China, male offspring are traditionally regarded as the primary heirs of household assets, particularly land. This study examines the impact of offspring gender composition on long-term agricultural investment behavior, using commercial organic fertilizer application as an example of a strategic long-term investment in farmland. Based on cross-sectional data from 4090 rice farming households across 10 cities (counties) in Hubei province, collected between 2021 and 2023, this analysis identifies three key findings. First, the absence of male offspring significantly reduces long-term agricultural investments, a result that remains robust even when addressing potential endogeneity biases using instrumental variable techniques. Second, households without male heirs exhibit stronger present-oriented preferences and diminished social capital, which further hinder long-term agricultural investments. Third, the negative impact of not having male offspring is more pronounced when the current agricultural decision-maker is male and when land marketization is underdeveloped. These findings underscore the complex interplay between gender norms and agricultural behavior, revealing significant socioeconomic implications of inheritance practices. The study provides insights into addressing these challenges by emphasizing the importance of promoting gender equality and advancing land marketization to enhance equitable land use and support long-term agricultural investment.

Mango is a delicious tropical fruit with high economic value worldwide. The Forest-Savanna Transition zone of Ghana contributes significantly to the production of mangoes for both local and international markets. The zone is plagued with the fruit fly ‘menace’ like all mango-producing areas in Ghana. Not much has been done in terms of species monitoring. A 24- month fruit fly monitoring survey was conducted in eight mango orchards, to assess the composition of fruit flies associated with Mango and their damage levels. Four para-pheromone attractants (Methyl Eugenol, Terpinyl Acetate, Trimedlure, and Cuelure) were used to monitor fruit flies. Eight improvised water bottle traps (two per lure) were purposefully deployed in each orchard. A total of 18 tephritid species belonging to five genera were recorded. Bactrocera (one species), Ceratitis (six species), Dacus (eight species), Zeugodacus (one species) and Xanthaciura (two species). Bactrocera dorsalis was the most abundant species (90% of the collected samples), while the native mango fly, C. cosyra constituted 0.5%. Dacus fuscovittatus and Dacus pleuralis were for the first time captured and identified in Ghana. Dacus langi, Dacus carnesi, Dacus (diastatus?), Ceratitis silvestrii and C. quinaria were recorded for the first time in the zone. The zone recorded a diversity index of 0.41. Damage levels ranged from 41–91%. Ten out of the 18 species, are of economic importance on mango and must be watched. Periodic updates on seasonal fluctuations, species composition and new arrivals are key to the successful implementation of any management strategy.