Collatz Conjecture Calculator
Trace the full Collatz sequence for any starting integer and count the steps to reach 1. Perfect for exploring one of mathematics' most famous unsolved problems visually and numerically.
About this calculator
The Collatz conjecture applies a simple rule repeatedly: if the current value n is even, divide by 2 (n → n/2); if it is odd, multiply by 3 and add 1 (n → 3n+1). No matter what positive integer you begin with, the sequence always appears to reach 1 — though this has never been mathematically proven for all integers. The number of steps required to reach 1 is called the stopping time or total stopping time. The formula iterated is: n(t+1) = n(t)/2 if n(t) is even, else 3·n(t)+1, continuing until n = 1 or the iteration limit is reached. The sequence can grow dramatically before collapsing — for example, starting at 27 the sequence peaks at 9232 before descending to 1 after 111 steps.
How to use
Enter starting_number = 6 and max_iterations = 1000. The calculator applies the rule step by step: 6 → 3 (even: 6/2) → 10 (odd: 3×3+1) → 5 (even: 10/2) → 16 (odd: 3×5+1) → 8 → 4 → 2 → 1. The sequence takes 8 steps to reach 1. The output shows step count = 8, peak value = 16, and the full sequence [6, 3, 10, 5, 16, 8, 4, 2, 1] if sequence display is enabled.
Frequently asked questions
Has the Collatz conjecture ever been proven for all positive integers?
No — despite being easy to state, the Collatz conjecture remains one of the most famous unsolved problems in mathematics. It has been verified computationally for all integers up to at least 2^68 (roughly 295 quintillion), but no general proof exists. Mathematician Paul Erdős said 'Mathematics is not yet ready for such problems.' Research by Terence Tao in 2019 showed that almost all Collatz sequences eventually reach a value close to 1, which is the closest anyone has come to a full proof.
What is the stopping time of a number in the Collatz sequence?
The stopping time (or total stopping time) is the number of iterations required for the sequence starting at n to reach 1. It varies wildly and non-monotonically — a larger starting number does not necessarily have a longer stopping time. For example, 27 takes 111 steps and peaks at 9232, while 256 reaches 1 in just 8 steps because it is a power of 2 and halves every iteration. Studying stopping times reveals fractal-like patterns in the integers.
Why do mathematicians find the Collatz conjecture so difficult to prove?
The sequence mixes two operations — multiplication and division — in a way that resists standard number-theoretic tools. The sequence behaves almost randomly, making it impossible to find a clean structural argument. Traditional techniques like induction, modular arithmetic, and generating functions have all failed to crack the general case. The problem sits at the boundary between number theory and dynamical systems, and resolving it likely requires fundamentally new mathematics. That accessibility coupled with its depth makes it an enduring fascination for both amateurs and professionals.