THE BINARY GOLDBACH CONJECTURE

<p>The Goldbach Conjecture, one of the oldest problems in mathematics, has fascinated and inspired many mathematicians for ages. In 1742 German mathematician Christian Goldbach, in a letter addressed to Leonhard Euler, proposed a conjecture. The modern-day version of the Binary/Strong Goldbach conjecture asserts that: Every even integer greater than 2 can be written as the sum of two primes. The conjecture had been verified empirically up to 4 × 1018, its proof however remains elusive, which seems to confirm that:</p><p><em>Some problems in mathematics remain buried deep in the catacombs of slow progress ... mind stretching mysteries await to be discovered beyond the boundaries of former thought. Avery Carr (2013) </em></p><p>The research was aimed at exposition, of the intricate structure of the fabric of the Goldbach Conjecture problem. The research methodology explores several topics, before the definite proof of the Goldbach Conjecture can be presented. The Ternary Goldbach Conjecture Corollary follows the proof of the Binary Goldbach Conjecture as well as the representation of even numbers by the difference of two primes Corollary. The research demonstrates that the Goldbach Conjecture is a genuine arithmetical question.</p>

Analysis of Ideas Changing in the History of Mathematical Analysis

Analysis is a branch of mathematics that deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiation, and integration. In the history of mathematics, analysis is the first subject became epidemic, the development of analysis originated from the British mathematician and physicist, the Sir Isaac Newton, and the German mathematician, Gottfried Wilhelm Leibniz, who developed the theory of Calculus, with hundred-years developing, the modern analysis is now very ample and has widely applications, it has grown into an enormous and central field of mathematical research, with applications throughout the sciences and in areas such as finance, economics, and sociology. In this paper, we investigated in some detail with the changing of the ideas in mathematical analysis. By numerating historical facts and the mathematical ideas, we concluded the result that the ideas changing is because of the changing of the studying objects, the conclusion are studied detailly in the paper.

The Binary Goldbach Conjecture

The Goldbach conjecture, one of the oldest problems in mathematics, has fascinated and inspired many mathematicians for ages. In 1742 a German mathematician Christian Goldbach, in a letter addressed to Leonhard Euler proposed a conjecture. The modern day version of the Binary/Strong Goldbach conjecture asserts that: Every even integer greater than 2 can be written as the sum of two primes. The conjecture had been empirically verified up to 4× 10 18​​​​​, its proof however remains elusive, which seems to confirm that: Some problems in mathematics remain buried deep in the catacombs of slow progress ... mind stretching mysteries await to be discovered beyond the boundaries of former thought. Avery Carr (2013) The research was aimed at exposition, of the intricate structure of the fabric of the Goldbach Conjecture problem. The research methodogy explores a number of topics, before the definite proof of the Goldbach Conjecture can be presented. The Ternary Goldbach Conjecture Corollary follows the proof of the Binary Goldbach Conjecture as well as the Representation of even numbers by the difference of two primes Corollary. The research demonstrates that the Goldbach Conjecture is a genuine arithmetical question. ​​​

Splitting Goldbach’s Twin Prime Conjecture Asunder for Even Numbers

The Goldbach Conjecture remains one of the several unsolved mathematical problems today, along with the Twin Prime Conjecture. It is said to be one of the simplest mathematical problems to state yet the most difficult to prove. In this paper, the author explores a few even numbers about the problem, and, using them as his sample size, juxtaposes his ideas with those of others through literature review. The author explores a solution to the Goldbach conjecture which was put forward about 275 years ago, by a German mathematician, Christian Goldbach, who was a contemporary of the genius German mathematician, Leonhard Euler. Euler in a letter to Goldbach, had confessed at the time that unfortunately, he (Euler) could not prove Goldbach’s mathematical poser. Through fundamental analysis and critical thinking, this author attempts to share his thoughts on how to resolve this long-standing mathematical debacle. The methodology used is basic, experimental, and exploratory, with the use of inferences through recognition of number patterns. We hope that this paper will make a small contribution to the discourse on Goldbach’s conjecture, whose solution has eluded mathematicians for about 275 years.

Join the community, with Hilbert’s twenty-three problems

“Join the community, with Hilbert’s twenty-three problems” tells the story of German mathematician David Hilbert who, in 1900 at the International Congress of Mathematicians in Paris, presented twenty-three problems in a first serious effort by any mathematician to curate a list of important open problems across mathematical subfields. In a speech accompanying his offering, he challenged mathematicians to solve all twenty-three problems in the next century. This chapter offers highlights and explanations of some of the twenty-three problems, including the Continuum Hypothesis, the Riemann Hypothesis, and Kepler’s Sphere-Packing Conjecture. With his broad reach, Hilbert understood the value of articulating community-wide mathematical goals, as well as the unifying effect that keeping score might have. Mathematics students and enthusiasts learn that they are part of a community of “zealous and enthusiastic disciples” of mathematics. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Follow your curiosity, along a space-filling curve

“Follow your curiosity, along a space-filling curve” tells the story of Italian mathematician Giuseppe Peano’s quest for and discovery of a space-filling curve—a curve that completely fills a space such as a square—that most mathematicians and scientists at the time did not believe existed. For example, Isaac Newton, in his Philosophiae Naturalis Principia Mathematica, tried to ban space-filling curves. The discussion of space-filling curves is enhanced with numerous hand-drawn sketches showing how to construct German mathematician David Hilbert’s space-filling curve. Mathematics students and enthusiasts are encouraged to foster a Peano-like curiosity in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Infinity

The paradoxes of Zeno in antiquity might seem like sophisms, but, as it has turned out, they shed light on mathematical infinity and have led to many derivative ideas in mathematics and science. Zeno’s paradoxes portray movement in terms of discrete points on a number line, where a move from A to B is done in separate (discrete) steps. But the gap in between is continuous. So, to resolve the paradoxes a basic distinction between discrete and continuous is required—a gap that was filled with the calculus much later. The infinity concept became in the nineteenth century the basis for the discovery of new numbers—by the German mathematician Georg Cantor—which seemed at the time to be counterintuitive. This chapter looks at the paradox of infinity and at Cantor’s ingenious discoveries.

Professor of matematics Aurel Edmund Voss and University of Lithuania

Professor Aurel Edmund Voss was a German mathematician, best known for his contributions to application of geometry in natural science and mechanics. The collection of books of professor A.E. Voss was bought and used by Faculty of Matematics and Nature of Univerity of Lithuania. In 1940’s the collection of books of prof. A.E. Voss was transfered to Vilnius University. The aim of the work was to investigate collections of books of libraries of Lithuania and to present books of prof. A.E Voss to scientific society of Lithuania.

Euclides Roxo e o movimento de reforma do ensino de Matemática na década de 30

Mostra como as idéias do educador brasileiro Euclides Roxo, determinantes para a elaboração dos programas de Matemática das reformas Campos e Capanema, enquadram-se no movimento renovador da Escola Nova e seguem as idéias do matemático alemão Felix Klein sobre a modernização do ensino da Matemática. Euclides Roxo contrapõe à orientação geral do ensino de Matemática da época, caracterizado por uma apresentação seca, abstrata e lógica, uma proposta pedagógica que leva em conta os interesses do aluno e seu estágio de desenvolvimento cognitivo e enfatiza a intuição, além de contextualizar a Matemática, deixando o tratamento rigoroso do assunto para níveis mais avançados da aprendizagem. Palavras-chave: história da educação no Brasil; história do ensino de Matemática; Euclides Roxo. Abstract The purpose of this paper is to show how the ideas of the Brazilian educator Euclides Roxo, which were decisive for the educational reforms made by Francisco Campos and Gustavo Capanema belong to the "Escola Nova" movement and also follow the ideas of the German mathematician Felix Klein about the modernization of mathematics teaching. Euclides Roxo, contrary to the accepted mathematics teaching methods of his time, which were formal abstract and logic, proposes to take in account the students' interests, their cognitive level and stresses intuition and contextualization, postponing a rigorous treatment of the subject to more advanced school levels. Keywords: history of education in Brazil; history of mathematics teaching; Euclides Roxo.

Between Berlin and Cambridge: classical conceptions of the general economic equilibrium in the late 1920s

This paper deals with some contributions to the debate on General Economic Equilibrium between the two world wars. Originating in Cambridge and Berlin, they differed from the Viennese contributions of the Walrasian perspective traditionally considered by the literature. They can be defined to represent the classical approach to general equilibrium. The authors considered are the Italian-born economist Piero Sraffa, the German mathematician Robert Remak and the Russian-born economist Wassily Leontief. The paper focuses in particular on the intellectual origins of their contributions in the Russian-German theoretical debate of the period 1890–1910 involving classical economists, Marx and Walras: these connections emerge clearly in the contributions of the German-Russian Mathematical School between the end of the nineteenth and the beginning of the twentieth centuries.