Os artigos de Euler sobre os números amigáveis

Among the many methodological resources that the mathematics teacher can use in the classroom, we can cite the History of Mathematics which has contributed to the development of activities that promotes students curiosity about mathematics and its history. In this regard, the present dissertation aims to translate and analyze, mathematically and historically, the three works of Euler about amicable numbers that were writed during the Eighteenth century with the same title: De numeris amicabilibus. These works, despite being written in 1747 when Euler lived in Berlin, were published in different times and places. The first, published in 1747 in Nova Acta Eruditorum and which received the number E100 in the Eneström index, summarizes the historical context of amicable numbers, mentions the formula 2nxy & 2nz used by his precursors and presents a table containing thirty pairs of amicable numbers. The second work, E152, was published in 1750 in Opuscula varii argument. It is the result of a comprehensive review of Euler s research on amicable numbers which resulted in a catalog containing 61 pairs, a quantity which had never been achieved by any mathematician before Euler. Finally, the third work, E798, which was published in 1849 at the Opera postuma, was probably the first among the three works, to be written by Euler

Revitalização do Rio Golandim (RN/Brasil) após décadas de contaminação por efluentes industriais e domésticos: este rio está recuperado?

Environmental pollution causes the loss of the quality of aquatic resources and also affects the health of human beings. The Golandim River is located in São Gonçalo do Amarante (RN Brazil) and had its water recovered seven years ago by measurements of parameters of the water s quality analyses physicochemistry, microbiological and heavy metals. However, it is not well established if this river is truly recovered, so this study provides a wide knowledge about the quality of these waters. Therefore, this investigation was accomplished by assays of ecotoxicology utilizing Ceriodaphnia dubia and of genotoxicity of the river s water using a biomarker Tradescantia Pallida (Trad-MCN). In set, it carried through a study of environmental perception through questionnaires that approached questions related to the profile of the interviewed one, knowledge on the environment and of the river Golandim with the community that lives in the neighborhoods of the river to diagnosis as they perceive the environment where live and its problems. The assays of the water had been carried out by collecting samples in three different sites of the Golandim River. They were collected between the periods of December 2010 (dry season in the northeast of Brazil) and July 2011 (rainy season in the northeast of). The analysis of the data allowed observing that the majority of the inhabitants are adult and presents a global vision of what it is part of the environment, the majority mentions the pollution of the river as one of the problems of the city, considering it serious. The ecotoxicology assay showed that there was not acute toxicity in all three samples collected. Meanwhile, all these three samples demonstrated significant chronic toxicity. The results from the Trad-MCN assay presented an increase in the frequency of micronucleus in one of the sites analyzed (S3) (p<0.01), in both seasons collected. On the other hand, the sites S1 and S2 did not presented a significant increase of micronucleus using this bioassay. The analyses of chemicals detected an increase in the levels of some metals, in different seasons and samples, which can be associates with some compounds found in urban and industrial areas. On the other hand, the physicochemistries parameters demonstrated that the Golandim River is recouped, when compared with the values presented at the CONAMA s legislation. However, these results indicate the presence of compounds capable of inducing chromosomal mutation in plants. On the other hand, the parameters physicistchemistries demonstrate that the river Golandim is if recouping, since when compares the values observed with the legislation of the CONAMA. All these results point to the fact that the Trad-MCN assay was sensitive and efficient biomarker for chromosomal instability and the C. dubia ecotoxicology assay was as though an efficient biomarker of toxicity of water s quality. The results from Trad-MCN associated with the ecotoxicology demonstrates that these analyses are important for environmental monitoring, once the first bioassay described above indicates alterations at the standards of cells and the other one indicates alterations at the standards of organisms. This study alerts for the necessity to carry out biological assays for the analyses of the water s quality