Sobre la Conjetura de Collatz

Abstract

The Collatz Conjecture, also known as 3x + 1, may seem like perhaps one of the problems simplest of mathematics. His statement is quite natural to understand: if you take an even number, this is divided by two; if the number is odd, it is multiplied by three and add 1 to the result. This process is carried out, iteratively, with the numbers obtained after each He passed. The conjecture states that no matter what number is taken, it always returns to two or to one, becoming absorbing states. For example, if you take the number 11 you get the following sequence of numbers 11; 34; 17; 52; 26; 13; 40; 20; 10; 5; 16; 8; 4; 2; 1, until the number 2 and later to 1; in fact, after reaching 1 this when iterating it becomes 4, then 2 and again it will reach 1, establishing a loop. What is most attractive is that this problem has been approached by large mathematicians such as Paul Erdos, who referred to this situation thus, “this is an extraordinarily difficult problem. difficult, completely beyond the reach of current mathematics, ”even Erdos offered $ 500 lares for the solution, as stated by Jeffrey Lagarias, who makes a rigorous study of the problem, analyzing it decade by decade, compiling works and attempts by other mathematicians to prove it. This paper studies the history of the Collatz Conjecture, its connection with other branches of mathematics, through the brief presentation of the most recent advances and discoveries in around this, as well as the graphical representation of the Collatz fractal and the iterations through computational methods which have made it possible to demonstrate its veracity with numbers up to 260, allowing a better understanding of the problem.