Syracuse Conjecture Quadrature
di Rolando ZucchiniAfter about 2300 years (Circle Quadrature; Archimèdès, Syracuse 287 – 212 BC) the history of mathematics repeats itself in a different problem
Abstract
After circa 2300 years (Archimedes, Syracuse 287 – 212 BC) the history of mathematics repeats itself in a different problem. In this paper is shown the true nature of Syracuse conjecture, tackling it from a completely dissimilar point of view than many previous attempts. One of its features suggests a process that leads to Theorem 2n+1, whose demonstration subdivided the set of odd numbers in seven disjoint 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. With Theorem of Independence we can manage the cycles of links to our liking, also to reach very high main horizons and when we decide go back to lower horizons. In this article it’s proved that Collatz conjecture is not fully verifiable. 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 special 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.
