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In this paper, we propose an elementary proof of Niven's Theorem in which the tangent function will have a primary role. [ABSTRACT FROM AUTHOR]

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of theElementsof Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann inAusdehnungslehrein 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying. [ABSTRACT FROM PUBLISHER]

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequences. [ABSTRACT FROM AUTHOR]