Collatz Conjecture Calculator

The Collatz Function

The Collatz function is a piecewise function that operates on positive integers. It follows a simple rule: if the number is even, divide it by 2; if the number is odd, multiply it by 3 and add 1. This creates a deterministic sequence that always moves toward 1, though the path can be surprisingly complex.

f(n) = \begincases n/2 & \textif n \text is even \\ 3n+1 & \textif n \text is odd \endcases

The Conjecture Statement

The Collatz conjecture states that for any positive integer n, the sequence n, f(n), f(f(n)), f(f(f(n))), ... will eventually reach 1. Once it reaches 1, it enters the cycle 1 → 4 → 2 → 1, which continues indefinitely. The conjecture claims this is true for ALL positive integers, making it a universal statement about the behavior of this function.

\forall n \in \mathbbN, \exists k \in \mathbbN : f^k(n) = 1

Stopping Time and Maximum Value

The stopping time σ(n) is the number of steps needed to reach 1 from n. The maximum value reached during the sequence can be dramatically larger than the starting number, creating "flights" that soar to great heights before eventually descending to 1. Some sequences reach values thousands of times larger than their starting point.

\textStopping time: \sigma(n) = \min\k : f^k(n) = 1\ \\ \textMaximum value: M(n) = \max\f^i(n) : i ≥ 0\

Statistical Properties

Empirical studies reveal fascinating statistical patterns. The average stopping time appears to grow logarithmically with n, while the distribution of stopping times shows complex, fractal-like structures. The ratio of even to odd steps tends toward 2:1, reflecting the fact that even numbers are divided by 2 while odd numbers are multiplied by 3.

\textAverage stopping time: \langle \sigma(n) \rangle \sim \log n \\ \textEven/odd ratio: \frac\texteven steps\textodd steps ≈ 2

Computational Verification

The conjecture has been computationally verified for all numbers up to 2^68 (approximately 3×10^20) using distributed computing. This represents an enormous computational effort, but it still doesn't constitute a mathematical proof. The verification continues with even larger numbers, but no counterexample has ever been found.

\textVerified for n < 2^68 ≈ 3 × 10^20 \\ \textNo counterexamples found

Mathematical Approaches

Various mathematical approaches have been attempted, including probabilistic methods, ergodic theory, and dynamical systems. The problem is related to the density of powers of 2 in the sequence and the distribution of odd numbers. Some researchers have shown that "almost all" sequences reach 1, but proving it for ALL sequences remains elusive.

\textProbabilistic approach: P(\sigma(n) < ∞) = 1 \text for almost all n

Generalizations and Variations

The Collatz problem has inspired numerous generalizations, including the 5n+1 problem, 3n-1 problem, and other linear functions. These variations often exhibit different behaviors - some may have multiple cycles or divergent sequences. The original 3n+1 problem remains the most studied due to its apparent simplicity and universal convergence.

\textGeneralized function: f_a,b(n) = \begincases n/2 & \textif n \text is even \\ an + b & \textif n \text is odd \endcases

Connection to Other Fields

The Collatz conjecture connects to many areas of mathematics: dynamical systems (as a discrete dynamical system), number theory (through arithmetic properties), graph theory (as a directed graph), and even theoretical computer science (as a halting problem). Its study has led to advances in understanding chaotic behavior in simple systems.

\textAs dynamical system: x_n+1 = f(x_n) \\ \textAs graph: G = (\mathbbN, \(n, f(n)) : n \in \mathbbN\)

Famous Sequences and Records

Certain starting numbers produce particularly interesting sequences. For example, starting with 27 requires 111 steps to reach 1, reaching a maximum value of 9,232. The number 97,571 requires 261 steps. These "high-flying" sequences demonstrate the unpredictable nature of the problem and why it's so difficult to prove.

\sigma(27) = 111, \quad M(27) = 9232 \\ \sigma(97571) = 261

Current Research and Open Problems

Current research focuses on partial results, such as proving the conjecture for specific classes of numbers or finding bounds on stopping times. The problem remains one of the most famous unsolved problems in mathematics, with a $1,000,000 prize offered by the Clay Mathematics Institute for its solution. The simplicity of the problem statement contrasts sharply with the depth of mathematical insight required to solve it.

\textOpen: Prove \forall n \in \mathbbN, \sigma(n) < ∞ \\ \textPrize: \$1,000,000 \text (Clay Mathematics Institute)

Worked Example

Inputs

• startNumber: 27
• maxSteps: 150

Result: Sequence reaches 1 in 111 steps with maximum value 9,232

Explanation

Starting with 27: The sequence takes 111 steps to reach 1, reaching a maximum value of 9,232 at step 23. This demonstrates the "high-flying" nature of some Collatz sequences, where the value can soar to great heights before eventually descending to 1. The sequence shows the characteristic pattern of dramatic ups and downs that make the Collatz conjecture so fascinating.

Second Scenario

Inputs

• startNumber: 34.75
• maxSteps: 150

Result: Sequence reaches 1 in 111 steps with maximum value 9,232

Explanation

This scenario uses different inputs (startNumber = 34.75, maxSteps = 150) to show how changing one variable affects the collatz conjecture result. Run the calculator above with these values to get the exact updated output with step-by-step work.