Collatz Conjecture Calculator

The Collatz conjecture states that for any positive integer n, repeatedly applying the rule (if even, divide by 2; if odd, multiply by 3 and add 1) will eventually reach 1.

No, it remains one of the most famous unsolved problems in mathematics. It has been verified computationally for all numbers up to 2^68.

Because the rule for odd numbers is to compute 3n + 1, which is the key operation that makes the problem interesting.

For numbers under 10 billion, the record is 2,091,132 steps starting from n = 6,706,172,790.

This is exactly what the conjecture claims, but it has not been proven. Every number tested so far does reach 1.

Hailstone sequence is another name for the Collatz sequence, because the numbers bounce up and down like hailstones in a cloud before falling to 1.

German mathematician Lothar Collatz proposed it in 1937. It is also known as the Ulam conjecture, Kakutani's problem, or the Syracuse problem.

The standard Collatz conjecture applies to positive integers. For negative numbers, sequences can enter cycles other than the 4—2—1 cycle.

The maximum value (called the "peak") can be much larger than the starting number. For n = 27, the peak is 9232.

No known pattern exists. The sequences appear pseudorandom, which is part of why the conjecture is so difficult to prove.

The number of steps varies dramatically. Small numbers can take just a few steps, while some numbers take millions of steps.

Starting with 1: 1 is odd — 3(1)+1 = 4 — 4 is even — 2 — 2 is even — 1. The sequence cycles: 1—4—2—1.

Yes, the Collatz function exhibits chaotic behavior. Small changes in starting values lead to dramatically different sequences.

Yes. You can plot the sequence values over step number to see the characteristic up-and-down pattern.

The conjecture has been verified using C, Python, Java, and many other languages, typically using parallel computing for large numbers.

Once the sequence reaches 1, it enters the cycle 1—4—2—1. This is the only known cycle for positive integers.

It has been shown that the Collatz problem is Turing-complete in a certain sense, meaning it can simulate any computation.

Given the extensive computational verification and statistical evidence, most mathematicians believe it is true, but proof remains elusive.

Yes, including the 5n+1 problem, the 7n+1 problem, and generalizations to other number systems and algebraic structures.

The total stopping time is the number of steps required for a Collatz sequence to reach 1 for the first time.

The Collatz Conjecture Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Collatz Conjecture Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.