Value of the golden ratio (number Phi) knowing the side length of a square

This paper explains how to obtain the number phi using a square with side length equal to a, the right triangle with sides a=2 and a, and a circle with radius equal to the hypotenuse of this right triangle. In particular, from a square whose side length is equal to a, we will show how to obtain a segment b in such a way that the value of a=b is the number phi. It is well known that this ratio is also calculated from equating the ratios obtained by dividing a segment of length a + b by a (being a always the largest segment) and a by b, that is, (a + b)=a = a=b. This equality is a consequence of the ratio of proportionality in triangles applying Thales’s Theorem. And, we must mention also how this golden ratio it is obtained as a consequence of the Fibonacci sequence. However, the golden ratio as a consequence of the limit of Fibonacci sequence was found later than many masterpieces, as for instance the ones of Leonardo da Vinci. This is the main reason because we analyzed how to find the proportionality golden ratio using the most common geometric figures and its symmetries. This paper aims to show how the golden ratio can be obtained knowing the side length a of a square

Analyzing Riemann's hypothesis

In this paper we perform a detailed analysis of Riemann's hypothesis, dealing with the zeros of the analytically-extended zeta function. We use the functional equation $\zeta(s) = 2^{s}\pi^{s-1}\sin{(\displaystyle \pi s/2)}\Gamma(1-s)\zeta(1-s)$ for complex numbers $s$ such that $0<{\rm Re(s)}<1$, as well as reduction to the absurd in combination with a deep numerical analysis, to show that the real part of the non-trivial zeros of the Riemann zeta function is equal to $1/2$, to the best of our resources. This is done in two steps. First, we show what would happen if we assumed that the real part of $s$ has a value between $0$ and $1$ but different from $1/2$, arriving to a possible contradiction for the zeros. Second, assuming that there is no real value $y$ such that $\zeta\left(1/2 +yi \right)=0$, by applying the rules of logic to negate a quantifier and the corresponding Morgan's law we also arrive to a plausible contradiction. Finally, we analyze what conditions should be satisfied by $y \in \mathbb R$ such that $\zeta(\displaystyle 1/2 +yi)=0$. While these results are valid to the best of our numerical calculations, we do not observe and foresee any tendency for a change. Our findings open the way towards assessing the validity of Riemman's hypothesis from a fresh and new mathematical perspective.Comment: 11 pages, 3 figure

Analyzing the Collatz Conjecture Using the Mathematical Complete Induction Method

In this paper, we demonstrate the Collatz conjecture using the mathematical complete induction method. We show that this conjecture is satisfied for the first values of natural numbers, and in analyzing the sequence generated by odd numbers, we can deduce a formula for the general term of the Collatz sequence for any odd natural number n after several iterations. This formula is used in one case that we analyze using the mathematical complete induction method in the process of demonstrating the conjecture