The well-known $abc$-conjecture concerns triples $(a,b,c)$ of nonzero integers that are coprime and satisfy ${a+b+c=0}$. The strong n-conjecture is a generalisation to n summands where integer solutions of the equation ${a_1 + \cdots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any n. In this paper, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac 54$; for odd $n \geq 5$ we even achieve $\frac 53$. In particular, Ramaekers’ conjecture is false for every ${n \ge 5}$.

Authoritarian survival theories maintain that dictators distribute rents to elites who can control the masses. Yet, it is unclear how dictators choose beneficiary elites. We argue that elites centrally placed in their locality’s family network enjoy greater influence on other community members and, thus, are more likely to be co-opted through distribution. We test this argument by compiling a novel dataset of Paraguayan family networks that we link to families who illegally benefited from public land grants from 1954 to 2007. Using a difference-in-differences in reverse design, we find that local families with higher network centrality were more likely to receive these grants during the 1954–88 dictatorship. We also show more affiliations with the ruling Colorado Party and incidents of repression—indicators of social control—in localities with more central families before 1989. Our work shows that family ties can serve to build authoritarian ruling coalitions.

Previous observational studies suggested that vitamin D may control the absorption of iron (Fe) by inhibition of hepcidin, but the causal relevance of these associations is uncertain. Using placebo-controlled randomisation, we assessed the effects of supplementation with vitamin D on biochemical markers of Fe status and erythropoiesis in community-dwelling older people living in the UK. The BEST-D trial, designed to establish the optimum dose of vitamin D3 for future trials, had 305 participants, aged 65 years or older, randomly allocated to 4000 IU vitamin D3 (n 102), 2000 IU vitamin D3 (n 102) or matching placebo (n 101). We estimated the effect of vitamin D allocation on plasma levels of hepcidin, soluble transferrin receptor (sTfR), ferritin, Fe, transferrin, saturated transferrin (TSAT%) and the sTfR–ferritin index. Despite increases in 25-hydroxy-vitamin D, neither dose had significant effects on biochemical markers of Fe status or erythropoiesis. Geometric mean concentrations were similar in vitamin D3 arms v. placebo for hepcidin (20·7 [se 0·90] v. 20·5 [1·21] ng/ml), sTfR (0·69 [0·010] v. 0·70 [0·015] µg/ml), ferritin (97·1 [2·81] v. 97·8 [4·10] µg/l) and sTfR–ferritin ratio (0·36 [0·006] v. 0·36 [0·009]), respectively, while arithmetic mean levels were similar for Fe (16·7 [0·38] v. 17·3 [0·54] µmol/l), transferrin (2·56 [0·014] v. 2·60 [0·021] g/dl) and TSAT% (26·5 [0·60] v. 27·5 [0·85]). The proportions of participants with ferritin < 15 µg/l and TSAT < 16 % were unaltered by vitamin D3 suggesting that 12 months of daily supplementation with moderately high doses of vitamin D3 are unlikely to alter the Fe status of older adults.

This study concerns prehistoric amber networks in north-eastern Iberia, emphasizing its distinct exchange dynamics compared to other regions of the Iberian Peninsula. Baltic amber dominated assemblages in this area from the Late Neolithic to the Bronze Age, contrasting with the prevalence of Sicilian amber in southern Iberia, or Cretaceous Iberian amber in the northern region. The findings underscore the region’s connection to southern France, with the Pyrenees serving as a cultural conduit, unlike the river Ebro, which acted as a boundary. Here the authors present the results of a Fourier transform infrared spectrometry (FTIR) analysis of twenty-one amber beads, primarily from collective burials. Eighteen were made of Baltic succinite. Baltic amber may have begun to arrive as early as 3634–3363 cal bc, and continued to be used until the Late Bronze Age. Exceptions included a unique spacer-bead made of gum and two bolus pigments misidentified as amber. The results highlight Iberia’s regional diversity in raw material sourcing and exchange, reflecting distinct sociocultural dynamics and challenging linear narratives of Iberian prehistory.

In the latter half of the eighteenth century, British civil engineers strove to enhance their status and assert the identity of their developing profession. Alongside associational and visual cultures, one means of achieving a sense of community was through the formation of a shared literary culture. As a profession notorious for what Torrens described as ‘papyrophobia’, it is perhaps surprising that many engineers, in this period, read widely and wrote extensively. John Smeaton (1724–92), for example, valued good authorship and experimented widely with literary form. James Brindley (1716–72), his contemporary, wrote sparingly, but nevertheless generated a literary strategy in support of his projects. Other engineers, such as John Phillips (fl. 1785–1813), made use of their engineering background and of engineering literature to create alternative careers. By exploring how mid- to late eighteenth-century engineers wrote, in order to persuade and to educate others as well as to publicize, record and defend their professional decisions, this paper will show how their reputations were dependent on literary constructions as much as on physical ones.

This article demonstrates that Ovid placed an incomplete reverse acrostic at Rem. am. 681–5 reading desin-. I will argue that it is intentional, noting that it fits the context in which it appears. Additionally, I will discuss how Ovid is drawing the reader’s attention to his engagement with the poetry of Catullus by referencing another possible acrostic in Catullus 36, as well as by playing with themes (and acrostics) from Catullus 8.