Random article ( of 1126 ) Latest updates

User Tools

Site Tools


content / mathematics / collatz_conjecture

Wikenigma - an Encyclopedia of Unknowns Wikenigma - an Encyclopedia of the Unknown

Collatz conjecture

Using the formula :

$$ {\displaystyle f(n)={\begin{cases}{\frac {n}{2}}\quad\quad\quad{\text{if }}n\equiv 0{\pmod {2}}\\[4px]3n+1\quad{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}} $$

The Collatz conjecture states that this process will eventually reach the number 1, regardless of which positive integer is chosen initially.

For example, using n = 12 generates the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.

The conjecture remains open : i.e. neither proven or disproven.

Further reading : Wikipedia


Also see : Juggler sequencesplugin-autotooltip__plain plugin-autotooltip_bigJuggler sequences

Juggler sequences were first proposed by American mathematician and author Clifford A. Pickover. They take the form :

$${\displaystyle a_{k+1}={\begin{cases}\left\lfloor a_{k}^{\frac {1}{2}}\right\rfloor {\mbox{if }}a_{k}{\mbox{ is even}}\\\\\left\lfloor a_{k}^{\frac {3}{2}}\right\rfloor {\mbox{if }}a_{k}{\mbox{ is odd}}.\end{cases}}}$$

It's conjectured that all Juggler Sequences eventually reach 1. The sequences have been verified for initial terms up to 106 - but the conj…

THIS WEBSITE DOES NOT USE TRACKING, ADVERTISING, OR ANALYTICAL COOKIES OF ANY KIND.
All essential cookies (for login status etc) are automatically deleted at the end of the session.
(full details here)

Show another (random) article

Suggestions for corrections and ideas for articles are welcomed : Get in touch!


Further resources :