Collatz Conjecture Explorer
Explore the famous 3n+1 problem by generating Collatz sequences and analyzing their behavior.
Enter a positive integer to generate its Collatz sequence (up to 15 digits)
The Collatz conjecture, also known as the 3n+1 problem, is one of the most famous unsolved problems in mathematics. Starting with any positive integer n, the sequence is generated by repeatedly applying a simple rule: if n is even, divide it by 2; if n is odd, multiply it by 3 and add 1.
The Rules: For any positive integer n: if n is even, the next term is n/2; if n is odd, the next term is 3n+1. The conjecture states that no matter what starting number you choose, the sequence will always eventually reach 1.
Example: Starting with 7: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1. This takes 16 steps and reaches a maximum value of 52.
The Mystery: Despite its simple definition, no one has been able to prove that every positive integer eventually reaches 1. The conjecture has been verified computationally for all numbers up to approximately 2^68, but a general proof remains elusive. Mathematician Paul Erdős famously said, "Mathematics is not yet ready for such problems."
Notable Sequences: The number 27 is famous for its surprisingly long sequence of 111 steps, reaching a maximum value of 9,232 before eventually descending to 1. This demonstrates how unpredictable the sequence behavior can be.
What is the Collatz conjecture?
The Collatz conjecture (also called the 3n+1 problem) states that starting from any positive integer, repeatedly applying the rule 'if even, divide by 2; if odd, multiply by 3 and add 1' will eventually reach 1. Despite its simple statement, it remains unproven after 80+ years.
Has the Collatz conjecture been proven?
No, the Collatz conjecture remains one of mathematics' most famous unsolved problems. It has been verified computationally for all numbers up to approximately 2^68 (about 295 quintillion), but no general proof exists. Paul Erdos said 'Mathematics is not yet ready for such problems.'
Why is the number 27 special in the Collatz sequence?
The number 27 is famous for its unexpectedly long Collatz sequence. Despite being a small number, it takes 111 steps to reach 1 and climbs to a maximum value of 9,232. This demonstrates how unpredictable the sequence behavior can be - nearby numbers often have vastly different sequence lengths.
What patterns exist in Collatz sequences?
While no complete pattern is known, some observations exist: powers of 2 reach 1 quickly, numbers of the form 4n+1 often grow before shrinking, and most sequences show a general downward trend despite local increases. The study of these patterns continues to fascinate mathematicians.