Collatz conjecture

(notes by Roberto Bigoni)

Collatz sequence


A number c0 ≥ 1 generates a Collatz's sequence C if it is the first element of a sequence of natural numbers ci such that


If C contains the number 1, this number can be considered the last of the sequence, because from this number forward the sequence oscillates in the cycle 1-4-2-1.

A conjecture is made that from whatever c0 a Collatz sequence starts, it reaches always 1.

Even though this conjecture has been tested for many numbers by calculation programs like the in the above form, it is not generally demonstrated for every natural number.

If your browser can run Java applets, you can visit the page CollatzJava, where a java applet calculates the sequences in wider, faster and more reliable way.

Many mathematicians were involved in the search of a demonstration. See:

last reviewed: July 2015