Juan Carlos Riano-Rojas

arXiv2024

An optimal convergent Collatz algorithm

In this research, an optimal algorithm for the Collatz conjecture is presented. Properties such as the convergence of the algorithm and an equation that relates the algorithm to the classical Collatz conjecture are obtained. It is validated that the proposed theory is correct with several examples; as a sector of mathematicians believes that the Collatz conjecture fails in some power of $3$. The algorithm was applied to the first $600$ powers of $3$, where the convergence of the proposed algorithm was verified.

math.GM↗

arXiv2023

Is it possible that the Goldbach's and Twins primes conjectures are true with an algebraic approach?

In this paper, using an algebraic approach, it is intended to show that the Goldbach's and Twin primes conjectures are true, building, for each $m>2$, an isomorphism between posets. One of the posets is the set of coprimes less than $m$, while the other is endowed with an operation that grants it an abelian group structure. Special features of this operation are demonstrated in the document, which allow characterizing the even numbers, as if they were their fingerprint; furthermore, such an operation locates, in a natural way, the pairs that satisfy the conjecture. Moreover, an algorithm that generates pairs of numbers that satisfy the conjecture is presented, and examples of some of the beautiful symmetries of the orbits of cyclic subgroups are shown, for the proposed abelian group.

math.GM↗