Collatz Conjecture Explained: Formula, Sequences, Properties & Examples
The Collatz Conjecture — also known as the 3n+1 problem, the Ulam Conjecture, or the Hailstone problem — stands as one of the most deceptively simple and stubbornly unsolved puzzles in all of mathematics. First posed by German mathematician Lothar Collatz in 1937, it has since captivated amateurs, professionals, and entire research communities for nearly a century. The conjecture is easy enough that a schoolchild can understand it, yet formidable enough that no mathematician has been able to prove or disprove it for all positive integers.
The Collatz Conjecture: Core Definition
The conjecture states the following: take any positive integer n. If n is even, divide it by 2. If n is odd, multiply it by 3 and add 1. Repeat this process indefinitely. The Collatz Conjecture asserts that no matter which positive integer you begin with, the sequence will always eventually reach the number 1. Once it reaches 1, the sequence enters the trivial cycle 1→4→2→1 and loops forever.
The Collatz Formula
Formally, the Collatz function is defined as: f(n) = n/2 when n is even, and f(n) = 3n + 1 when n is odd. Applying this function repeatedly produces what is called the Collatz sequence or hailstone sequence for that starting number. The stopping time (or total stopping time) of n is the number of function applications required to first reach 1.
Collatz Sequence Examples
Understanding the conjecture is easiest through concrete examples. Starting from n = 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1 (8 steps). Starting from n = 27, the sequence rises dramatically to a peak value of 9,232 before eventually descending to 1 after 111 steps — a striking illustration of how unpredictable these sequences can be even for small starting values.
Example 1 — n = 6 (Stopping time: 8 steps)
6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Example 2 — n = 27 (Stopping time: 111 steps, Peak: 9,232)
27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → … → 9232 → … → 1
Example 3 — n = 837799 (Record stopping time: 524 steps)
837799 → 2513398 → 1256699 → … → 1
Properties of the Collatz Sequence
While no complete mathematical characterisation exists, several properties of Collatz sequences are well established. Powers of 2 are the simplest cases — for n = 2^k, the sequence simply halves k times to reach 1, with a stopping time of k. Odd numbers always increase immediately (since 3n+1 is always even when n is odd), which is why sequences tend to show large spikes early. The sequence for n = 1 is trivially: 1 → 4 → 2 → 1, a 3-cycle.
Usage and Significance in Mathematics
The Collatz problem is used in mathematics education to introduce concepts such as iteration, number theory, and computational verification. In computer science, it is a classic benchmark for recursive and iterative algorithm design. Researchers have used the Collatz problem to explore connections with ergodic theory, dynamical systems, and even aspects of number-theoretic randomness. Computational efforts have verified the conjecture for all integers up to at least 2^68 — roughly 295 quintillion — without finding a counterexample. Yet the general case remains open. The famous mathematician Paul Erdős remarked: "Mathematics is not yet ready for such problems." In 2019, Terence Tao proved that almost all Collatz orbits do eventually reach arbitrarily close to 1, representing the most significant progress to date — though still short of a complete proof.
How to Use the Collatz Conjecture Calculator
To use this tool, enter any positive integer in the Single Number calculator and click "Calculate Collatz Sequence." The results display the stopping time (number of steps to reach 1), the maximum value reached during the sequence, the number of even and odd steps, and a visual bar chart of the entire sequence path. For bulk analysis, switch to the Bulk Collatz tab, paste a list of positive integers (one per line), or upload a CSV/TXT file. Results can be copied to clipboard or downloaded as a CSV for further analysis in Excel or other tools.