k-generalized Fibonacci numbers of the form 1+2^{n_1}+4^{n_2}+cdots+(2^{k})^{n_k}

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence (F_n^{(k)})_{n>= 2-k} whose first k terms are 0, ..., 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, we investigate k-generalized Fibonacci numbers written in the form 1+2^{n_1}+4^{n_2}+cdots+(2^{k})^{n_k}, for non-negative integers n_i, with n_k >= max{ n_i | 1<= i <= k-1}

Luca Pacioli and his 1500 book de Viribus Quantitatis

Tese de mestrado, História e Filosofia das Ciências, Universidade de Lisboa, Faculdade de Ciências, 2015As the field grows, History of Science has become wider-ranging than a purely progress-oriented view of the history of Science. The History of Mathematics, even though more resilient, has shown to follow the same development. The present dissertation tries to contribute to the general study by shedding some light on a book which has been belittled, misinterpreted or ignored altogether, De Viribus Quantitatis, one of the major historical recreational mathematics books, and its author Luca Pacioli. This text aims to provide a modern updated survey of the content of this book for related studies, as well as a résumé of its contents.Com o crescimento do ramo de História das Ciências este tem vindo a desenvolver um olhar mais abrangente que a clássica visão dedicada ao progresso das ideias científicas. A História da Matemática, embora mais resiliente, também tem vindo a mostrar interesse em expandir os seus horizontes. No presente texto tentamos contribuir para o estudo geral destas disciplinas estudando um tratado que pouca atenção tem tido até ao momento, sendo até mesmo mal interpretado. Trata-se De Viribus Quantitatis, sendo este um dos maiores compêndio de matemática recreativa no seu contexto histórico. O seu autor, Luca Pacioli, sendo uma personalidade de grande interesse e mais conhecido por outras obras suas. Nestas páginas tentamos fornecer uma versão atualisada da documentação relativa ao tratado tal como um resumo dos seus conteúdos

A Link to the Math. Connections Between Number Theory and Other Mathematical Topics

Number theory is one of the oldest mathematical areas. This is perhaps one of the reasons why there are many connections between number theory and other areas inside mathematics. This thesis is devoted to some of those connections. In the first part of this thesis I describe known connections between number theory and twelve other areas, namely analysis, sequences, applied mathematics (i.e., probability theory and numerical mathematics), topology, graph theory, linear algebra, geometry, algebra, differential geometry, complex analysis, physics and computer science, and algebraic geometry. We will see that the concepts will not only connect number theory with these areas but also yield connections among themselves. In the second part I present some new results in four topics connecting number theory with computer science, graph theory, algebra, and linear algebra and analysis, respectively. [...] In the next topic I determine the neighbourhood of the neighourhood of vertices in some special graphs. This problem can be formulated with generators of subgroups in abelian groups and is a direct generalization of a corresponding result for cyclic groups. In the third chapter I determine the number of solutions of some linear equations over factor rings of principal ideal domains R. In the case R = Z this can be used to bound sums appearing in the circle method. Lastly I investigate the puzzle “Lights Out” as well as variants of it. Of special interest is the question of complete solvability, i.e., those cases in which all starting boards are solvable. I will use various number theoretical tools to give a criterion for complete solvability depending on the board size modulo 30 and show how this puzzle relates to algebraic number theory

Current Trends in Symmetric Polynomials with their Applications

This Special Issue presents research papers on various topics within many different branches of mathematics, applied mathematics, and mathematical physics. Each paper presents mathematical theories, methods, and their application based on current and recently developed symmetric polynomials. Also, each one aims to provide the full understanding of current research problems, theories, and applications on the chosen topics and includes the most recent advances made in the area of symmetric functions and polynomials

Automated theory formation in pure mathematics

The automation of specific mathematical tasks such as theorem proving and algebraic manipulation have been much researched. However, there have only been a few isolated attempts to automate the whole theory formation process. Such a process involves forming new concepts, performing calculations, making conjectures, proving theorems and finding counterexamples. Previous programs which perform theory formation are limited in their functionality and their generality. We introduce the HR program which implements a new model for theory formation. This model involves a cycle of mathematical activity, whereby concepts are formed, conjectures about the concepts are made and attempts to settle the conjectures are undertaken.HR has seven general production rules for producing a new concept from old ones and employs a best first search by building new concepts from the most interesting old ones. To enable this, HR has various measures which estimate the interestingness of a concept. During concept formation, HR uses empirical evidence to suggest conjectures and employs the Otter theorem prover to attempt to prove a given conjecture. If this fails, HR will invoke the MACE model generator to attempt to disprove the conjecture by finding a counterexample. Information and new knowledge arising from the attempt to settle a conjecture is used to assess the concepts involved in the conjecture, which fuels the heuristic search and closes the cycle.The main aim of the project has been to develop our model of theory formation and to implement this in HR. To describe the project in the thesis, we first motivate the problem of automated theory formation and survey the literature in this area. We then discuss how HR invents concepts, makes and settles conjectures and how it assesses the concepts and conjectures to facilitate a heuristic search. We present results to evaluate HR in terms of the quality of the theories it produces and the effectiveness of its techniques. A secondary aim of the project has been to apply HR to mathematical discovery and we discuss how HR has successfully invented new concepts and conjectures in number theory