Syracuse Conjecture QuadratureMarchio Rolando Zucchini
After about 2300 years (Circle Quadrature; Archimèdès, Syracuse 287 – 212 BC) the history of mathematics repeats itself in a different problem
The conjecture of Syracuse, or Collatz conjecture, in this paper it is approached highlighting some of its features. One of them suggests a process that leads to Theorem 2n+1, whose demonstration subdivided the set of odd numbers in seven subsets which have different behaviors applying algorithm of Collatz. It allows us to replace the Collatz cycles with the cycles of links, transforming their oscillating sequences in monotone decreasing sequences. By Theorem of Independence we can manage cycles of links as we like, also to reach very high horizons and when we decide go back to lower horizons. In this article it’s proved that Collatz conjecture is not fully demonstrable. In fact, if we consider the banal link n < 2n, there are eight cycles which connect each other in an endless of possible links. It is a type of Circle Quadrature, but its statement is confirmed. In other words: BIG CRUNCH (go back to 1) is always possible, but BIG BANG (to move on) has no End.