Collatz Conjecture
Generate the complete Collatz (3n+1) sequence from any integer. See steps to 1, max value, stopping time, and full trajectory. Free online number theory tool.
About This Calculator
The Collatz Conjecture Calculator generates the complete sequence for any positive integer starting value following the rules of the famous 3n+1 problem: if the current number is even, divide by 2; if odd, multiply by 3 and add 1. The sequence continues until it reaches 1, as conjectured. This calculator shows every term, the total number of steps (stopping time), and the maximum value reached during the sequence.
The Collatz conjecture was first proposed by Lothar Collatz in 1937 and has been extensively studied by mathematicians including Paul Erdős, John Conway, and Terence Tao. Despite decades of research and computational verification up to 2^6^8, no proof has been found. It is considered one of the most famous unsolved problems in mathematics.
Example: Starting from n = 6: 6 (even) -> 3 (odd) -> 10 (even) -> 5 (odd) -> 16 (even) -> 8 -> 4 -> 2 -> 1. Total 8 steps. This calculator shows each intermediate value, making it easy to study the behavior of different starting numbers.
Frequently Asked Questions
What is the Collatz Conjecture?
The Collatz conjecture (also known as the 3n+1 problem) is an unsolved problem in mathematics. It states that for any positive integer n, repeatedly applying the rule -- if n is even, divide by 2; if n is odd, multiply by 3 and add 1 -- will eventually reach the number 1. Despite being tested for numbers up to 2^6^8, no counterexample has been found, but a proof remains elusive.
How many steps does a Collatz sequence take to reach 1?
The number of steps varies greatly. Small numbers like 4 take only 2 steps (4->2->1), while 27 takes 111 steps to reach 1, reaching a maximum value of 9232 along the way. There is no known formula to predict the stopping time, which is part of what makes the problem so intriguing.
Why is the Collatz Conjecture important in mathematics?
The Collatz conjecture is important because it is deceptively simple to state yet has resisted proof for over 80 years. It connects to dynamical systems, number theory, computability theory, and has been studied by Erdős, Conway, and Terence Tao. Paul Erdős famously said, "Mathematics is not yet ready for such problems."
What happens if the Collatz Conjecture is false?
If false, there exists a starting number that either diverges to infinity (grows without bound) or enters a cycle other than 4->2->1. Extensive computer searches have verified the conjecture for all numbers up to 2^6^8 (about 295 quintillion), but no counterexample has been found. A proof remains one of the great open problems.
Is the Collatz Conjecture related to the halting problem?
Yes, in 2015 it was shown that a generalization of the Collatz problem is computationally undecidable (equivalent to the halting problem). This means that for some variants of the 3n+1 rule, there is no algorithm that can determine whether a given starting number reaches 1 or not.
What is the highest number reached by a Collatz sequence?
The maximum value a Collatz sequence reaches is called the peak value. For n = 27, the peak is 9232. For n = 9663, the peak reaches 27,111,688. The peak tends to be much larger than the starting number for many inputs, which is why the problem is connected to dynamical systems and chaos theory.
How is the Collatz Conjecture taught in schools?
The Collatz conjecture is often introduced in mathematics enrichment programs and competition math across India, the US, and the UK as an accessible example of an unsolved problem. It helps students understand iteration, sequences, modular arithmetic, and the difference between conjectures and theorems.