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Journal articles on the topic 'Problems of Diophantus' "Arithmetica"'
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1
Shiu, Peter. "The gaps between sums of two squares." Mathematical Gazette 97, no. 539 (2013): 256–62. http://dx.doi.org/10.1017/s0025557200005842.
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Problems concerning the setof numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identitywhich shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.
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Acerbi, Fabio. "The meaning of πλασμαтιкόν in Diophantus’ Arithmetica". Archive for History of Exact Sciences 63, № 1 (2008): 5–31. http://dx.doi.org/10.1007/s00407-008-0028-8.
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Sasaki, Chikara. "D'al-Khwārizmī à Descartes." Arabic Sciences and Philosophy 23, no. 2 (2013): 319–25. http://dx.doi.org/10.1017/s0957423913000052.
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The volume D'al-Khwārizmī à Descartes is a monumental contribution to the world history of mathematical sciences, showing clearly that Arabic mathematics was an indispensable predecessor of early modern European mathematics. Roshdi Rashed is known, first of all, as an editor of classical mathematical writings in Arabic by such authors as al-Khwārizmī, Thābit ibn Qurra, Ibrāhīm ibn Sinān, Ibn al-Haytham, al-Khayyām, Sharaf al-Dīn al-Ṭūsī, as well as of the Arabic versions of Apollonius' Conics, Diophantus' Arithmetica, and Diocles' Burning Mirrors. As the volume under review shows, he is also a historian of mathematics of the first class who has transformed historiography. This book is, in a sense, a manifesto of Prof. Rashed's entire œuvre.
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Biase, Emanuele de. "The Vat. Gr. 191, Manuel Bryennius, and a circle of scholars alternative to that of Maximus Planudes." Scriptorium 70, no. 2 (2016): 349–63. http://dx.doi.org/10.3406/scrip.2016.4415.
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This article intends to give consistent answers to all the questions that a complex manuscript such as Vat. gr. 191 presents. All the descriptions hitherto published (and an autoptic examination) of this codex give a clear idea of the complexity of its genesis. After highlighting the inconsistencies found in the scientific contributions that aimed at identifying the revisor of this miscellany, this work presents all the information present in Vat. gr. 191 about the revisor, and then matches it with the profile of a prominent scholar of the time, Manuel Bryennius. This hypothesis finds further confirmation in the textual tradition of some of the texts contained in this manuscript, namely, Ptolemy’s Geographia and Diophantus’ Arithmetica.
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Cartiere, Carmelo R. "An Analytical Study of Diophantine Equations of Pythagorean Form: Causal Inferences on Hypothesized Relations between Quadratic and Non-quadratic Triples." Athens Journal of Education 12, no. 3 (2025): 527–46. https://doi.org/10.30958/aje.12-3-10.
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In XVII century, presumably between 1637 and 1638, with a note in the margin of Diophantus’ “Arithmetica”, Pierre de Fermat stated that Diophantine equations of the Pythagorean form, , have no integer solutions for , and . Of this statement, however, Fermat never provided a proof. Only after more than 350 years, in 1994, Prof. Andrew J. Wiles was finally successful in demonstrating it (Wiles, 1995; Taylor & Wiles, 1995; Boston, 2008). However, Wiles’ proof adopts calculus techniques far beyond Fermat’s knowledge. Our aim is to show an analytical method to attempt a proof to Fermat’s last theorem with the only use of elementary calculus techniques. Keywords: number theory, Diophantine equations, Pythagorean Theorem, Fermat’s last theorem, numerical analysis
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Saliba, George. "Les arithmetiques. Diophante, Roshdi RashedBooks IV to VII of Diophantus' Arithmetica in the Arabic Translation Attributed to Qusta ibn Luqa. Diophantus, Jacques Sesiano." Isis 79, no. 2 (1988): 266–70. http://dx.doi.org/10.1086/354702.
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Knorr, Wilbur, and Jacques Sesiano. "Books IV to VII of Diophantus' Arithmetica: In the Arabic Translation Attributed to Qusta Ibn Luqa." American Mathematical Monthly 92, no. 2 (1985): 150. http://dx.doi.org/10.2307/2322659.
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Hogendijk, Jan P. "Books IV to VII of Diophantus' Arithmetica in the Arabic translation attributed to Qusṫā ibn Lūqā". Historia Mathematica 12, № 1 (1985): 82–85. http://dx.doi.org/10.1016/0315-0860(85)90077-1.
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Knorr, Wilbur. "Books IV to VII of Diophantus' Arithmetica: In the Arabic Translation Attributed to Qusṭâ Ibn Lûqâ. By Jacques Sesiano". American Mathematical Monthly 92, № 2 (1985): 150–54. http://dx.doi.org/10.1080/00029890.1985.11971565.
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Borić, Marijana. "Antički korijeni Getaldićeva rada na razvoju matematičke analize i sinteze." Radovi Zavoda za hrvatsku povijest Filozofskoga fakulteta Sveučilišta u Zagrebu 52, no. 1 (2020): 95–120. http://dx.doi.org/10.17234/radovizhp.52.4.
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The diverse opus of Marin Getaldić can methodologically and conceptually be divided into two parts. Getaldić’s early works can be considered as a reinterpretation of selected works from the ancient Greek and Roman tradition with the aim of transmitting ancient knowledge and theories, but also as an effort to further explore and improve these teachings within the framework of ancient Greek mathematical methods. In his more mature works, Getaldić was focused on the problem of the method. For twenty years, in his native Dubrovnik, he was developing the ideas he had encountered on a study trip across Europe, independently and almost completely isolated from the intense developments in the European scientific community in the first decades of the 17th century. He summarized the findings of his research in a seminal, five-volume work De resolutione et compositione mathematica (Rome 1630). Although Getaldić operated in an environment that was permeated by the Renaissance and humanist influences, in his local environment knowledge was transferred more slowly than in Western European countries where modern science emerged during the 16th and 17th centuries. In Dubrovnik isolation, he created new theoretical and practical knowledge, as well as original works that echoed in the European scientific community not only during his lifetime, but also later, during the 17th and 18th centuries. His example shows that the transfer of knowledge did not take place only from European epistemological centers to the periphery, for it shows that the scientific transfers within Europe went in both directions. He worked at a time when the accumulated knowledge about ancient works and the spread of humanistic education outgrew the ancient tradition, and gradually, after the methodological transformation, modern science was founded and shaped. It took almost twenty centuries for the ancient mathematical methodology, complemented by knowledge assimilated from the Arab and Indian mathematical traditions, to be conceptually modified and new methods aiming at achieving new theoretical knowledge and practical solutions to be developed. In building his rich opus, Getaldić relied heavily on the original ancient mathematical methods, which he consistently applied to a variety of problems. His work was largely based on the works of Greek mathematicians, among whom Pappus and Diophantus stand out, and was influenced by Eudoxus’ theory of scale and Archimedes’ application of logical methodology, i.e. arithmetic interpretation of geometry. Getaldić combined different tendencies of ancient Greek mathematics in a unique and fruitful way. After mastering Viète’s symbolic algebra that operated with general quantities, Getaldić systematically explored the possibilities of symbolic algebra in relation to ancient mathematical methods, which played a crucial role in the further development of modern mathematics and gradually lead to another major conceptual change in mathematical history. The change did not only affect mathematics, but also enabled the emergence of new, simpler and more exact interpretations in other sciences as well.
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Zong, Qi. "Integer Solution for a Class of Indeterminate Equations." Journal of Physics: Conference Series 2386, no. 1 (2022): 012014. http://dx.doi.org/10.1088/1742-6596/2386/1/012014.
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Abstract The mathematical discipline of number theory has a long history. It originated in the ancient Greek period in natural mathematics and geometry. With the progress of science and technology, people began to recognize the importance of “number,” which led to the rapid development of number theory and gradually becoming an independent discipline. Indeterminate equations are one of the most important branches of number theory and have always received a lot of attention from the mathematical community. The problem of integer solutions of indeterminate equations was also called the Diophantus equation because of the profound study of these equations by Diophantus in the early third century. However, there is no unified method for solving the Diophantus equation today, so the investigation of its integer solution often needs to be combined with previous studies. This paper focuses on the application of Gaussian rings of integers and congruence theory to the study of integer solution problems for indeterminate equations of the form x^2+4^k=4y^n (k=1,2 n=3,5..). The study generalizes the understanding of the local distribution rules in specific situations, which helps to simplify the discussion of the classification of this category of problems and provides alternative ideas for subsequent more in-depth studies by others. This paper concludes that every such indeterminate equation can be discussed in x ≡ 1 (mod 2) and x ≡ 0 (mod 2). A template for discussion is provided for the first scenario, and only when 2k-1=n can there exist only an integer solution (x,y)=(±2^k,3) for the second scenario.
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Mastorakou, Stamatina. "2023 Books Received / Livres reçus." Aestimatio: Sources and Studies in the History of Science 4 (November 4, 2024): 1–2. http://dx.doi.org/10.33137/aestimatio.v4.43007.
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Please note that the editorial foreword does not include an abstract. Jeremy Armstrong War and Society in Early Rome. Cambridge: Cambridge University Press, 2021. Claire Bubb. Dissection in Classical Antiquity. Cambridge: Cambridge University Press, 2022. David H. Camden. The Cosmological Doctors of Classical Greece: First Principles in Early Greek Medicine. Cambridge: Cambridge University Press, 2023. Nathan Carlig ed. «Mes voeux les meilleurs et santé continuelle». Réponses aux épidémies dans le monde Gréco-Romain. Liège: Presses Universitaires de Liège, 2023. Jean Christianides and Jeffrrey Oaks. The Arithmetica of Diophantus. A Complete Translation and Commentary. Abingdon/New York: Routledge, 2023. Radcliffe G. Edmonds III. Drawing Down the Moon: Magic in the Ancient Greco-Roman World. Princeton/ Oxford: Princeton University Press, 2019. Ida Fröhlich ed. Science in Qumran Aramaic Texts. Tübingen: Mohr Siebek, 2022. Brian Glenney and José Filipe Silva edd. The Senses and the History of Philosophy. London/New York: Routledge, 2023. Matthias Heiduk, Klaus Herbers, and Hans-Christian Lehner edd. Prognostication in the Medieval World. Berlin/Boston: De Gruyter, 2020. Ralph Jackson. Greek and Roman Medicine at the British Museum: The Instruments and Accoutrements of Ancient Medicine. London: British Museum Press, 2023. J. Cale Johnson and Alessandro Stavru. Visualizing the Invisible with the Human Body: Physiognomy and Ekphrasis in the Ancient World. Boston: De Gruyter, 2020. Cécile Michel, and Karine Chemla edd. Mathematics, Administrative and Economic Activities in Ancient Worlds: Why The Sciences of the Ancient World Matter. Cham: Springer, 2020. Iulian Moga. Religious Excitement in Ancient Anatolia: Cult and Devotional Forms for Solar and Lunar Gods. Leuven/Paris/Bristol: Peeters, 2019. Courtney Ann Roby. The Mechanical Tradition of Hero of Alexandria.Cambridge: Cambridge University Press, 2023. Nathan Sidoli and R. S. D. Thomas. The Spherics of Theodosios. Scientific Writings from the Ancient and Medieval World. Abingdon/New York: Routledge, 2023. Virginia Trimble and David A. Weintraub edd. The Sky Is for Everyone: Women Astronomers in Their Own Words. Princeton: Princeton University Press, 2022. Benno van Dalen. Ptolemaic Tradition and Islamic Innovation: The Astronomical Tables of Kūshyār ibn Labban. Turnhout: Brepols, 2021. How to cite: Mastorakou, S. "2023 Books Received/Livres Reçus". Aestimatio: Sources and Studies in the History of Science (2023) 4: bm01 1–2. https://doi/10.33137/aestimatio.v4.43007
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Marcinowski, J., and M. Sadowski. "Using the ERFI Function in the Problem of the Shape Optimization of the Compressed Rod." International Journal of Applied Mechanics and Engineering 25, no. 2 (2020): 75–87. http://dx.doi.org/10.2478/ijame-2020-0021.
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AbstractThe shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was 80% and 117%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible.The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows.There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range.One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper Summa de arithmetica, geometria, proportioni et proportionalita (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. [1]).We can see, that already then, they were definitely important (complex numbers).
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Sverchevska, I. A. "A generalization of Diophantus' substitutions." February 16, 2020. https://doi.org/10.5281/zenodo.3669033.
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<em>The study considers the Diophantus' methods of solving certain systems of nonlinear equations given in the "Arithmetica" treatise, and further suggests the generalization of solutions founded by Diophantus.</em> <em>The author regards the method of generalization of mathematical theories and statements as a means of developing students' creative thinking. It formates their readiness to develop creative thinking in their future pupils.</em> <em>To archive this goal, the work proposes a historical approach based on using some facts from the history of mathematics, as well as famous mathematical problems. The latter include the problems from ancient treatises and the problems, formulated by famous mathematicians.</em> <em>The study investigates the methods of solving word problems authored by Diophantus of Alexandria (3rd century CE), the last great mathematician of Antiquity, in his "Arithmetica" treatise.</em> <em>The article considers the problems with mathematical models, based on systems of nonlinear algebraic equations containing fewer equations than unknowns. The work gives a generalization of Diophantus' approaches to solving some problems from "Arithmetica" treatise. The work derives formulas for an infinite number of solutions, including Diophantine solution.</em> <em>Generalized solutions are identified using the identity for the sum of the number and the square of half of the difference between denominator and quotient of this number.</em> <em>The solutions of the system are also presented as linear or quadratic functions with parameter-dependent coefficients. The study obtains the solutions for specific parameter values. Otherwise, the sufficient condition for free terms values is proven, in which all the solutions are integers.</em> <em>The paper concludes that the methods of solving historical tasks, as well as their generalization, ought to be an important component of training future teachers of mathematics.</em>
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Sidoli, Nathan. "The Arithmetica of Diophantus: A Complete Translation and Commentary by Jean Christianidis and Jeffrey Oaks." Aestimatio: Sources and Studies in the History of Science 4 (December 17, 2024). https://doi.org/10.33137/aestimatio.v4.44123.
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This book is not only an excellent translation and study of the extant Greek and Arabic sources for Diophantus’ Arithmetica, it is also an important piece of scholarship in the history of premodern mathematics. The historiographic significance of this book comes from both its argument for the place of the Arithmetica in our understanding of the history of algebra and from the methodologies that the authors employ in order to make their case.
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Kydyraliev, S. K., A. B. Urdaletova, and S. N. Dzhaparova. "THE KYRGYZ METHOD FOR THE DEVELOPMENT OF MATHEMATICS AND MODERNITY." Bulletin of Issyk-Kul University, July 30, 2024. http://dx.doi.org/10.69722/1694-8211-2024-57-28-34.
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The educational process consists of two main types of activities: the teaching activity of the teacher in educating students in knowledge, skills, and life experience, and the learning activity of the student in mastering the content taught by the teacher. Defining the learning process, teaching guides and determines learning. Therefore, the learning process is often explained as the process of educating the student. As a result, the learning process is perceived as a mechanism for the "transmission" of teaching content from teacher to student, with the teacher directly shaping the student. It formalizes the algorithm for solving problems − the problem-solving algorithm using the method of Diophantus al-Khwarizmi. This article presented a solution to text problems in algebra for grade 7 using the Kyrgyz method and a system of linear equations. It should be noted that this type of thinking was successfully used by many ancient Egyptian, Babylonian, Chinese and Indian mathematicians. The generalization of the described method can be successfully used to solve various problems.