List of articles

 

 

Collatz (3n+1, 3x+1)-problem

 

Besides some statistical extrapolations and some sportive records, there is still not much proven in the Collatz-problem, which comes up sometimes in sci.math, alt.math.recreational od de.sci.mathematik.

I did some intense studies in the last years, but didn't collect the results in internet-like articles. I'm going to do that occasionally. I produced some views into the collatz-tree; some as excel-sheet, some as graphics. A special nice graphic is a fractal like one, which orders the tree into a round brush-like scheme. See "about loops"

 

My 2004 studies concerned the question, whether a loop exists in the 3x+1-problem, or better, how to prove, that that doesn't exist. Several times I supposed, I succeeded, but the proofs were always insufficient (if not false). But they gave new insight in the problem in a way, that I didn't see in the internet anywhere till now. Even if the formulae are still not sufficient to disprove the 3x+1-loop, they illustrate some properties and explain, why a loop in 5x+1 is existent, or give a handy tool, to enumerate the loop candidates and disprove the existence by examination of a finite number of tries. See "about loops"


A more concise discussion of the m-cycle-problem is in "m-peak-cycles". An 1-peak-cycle is a cycle, where the elements of a compressed transformation ( n2 =T(n1 ; A) := n2 = (3*n1+1)/2^A where n1 and n2 are odd positive numbers, and A is chosen appropriately) are ascending and the cycle is closed by a single descending step, such that n = T(n; 1,1,1,1...,1,A) = PT(n1,N:A) where N-1 "1" occur, and 1 "A". An m-fold concatenated series of 1-peak-transformation, finally closed to form a cycle, is then a "m-peak-cycle":  n = PT(n; N1:A1,N2:A2,...Nm:Am).
See "m-peak-cycles"

 

 

A short discussion of  m-peak-cycles and certain bounds

last update: 10.8.2006       m-peak-cycles

 

About cycles / loops in the Collatz-problem
The article contains a short discussion of textual and graphical trees as well as some approximation-discussion of n*log(3) - m* log(2)

last update: 15.5.2004       About loops

 

 

 

Gottfried Helms

Univ Kassel