Collatz conjecture
Proof of Predictability
To prove the **predictability** of the Collatz conjecture, we demonstrate that for any starting number \( n \), we can predict when the sequence will produce a number smaller than \( n \). This involves analyzing the transformations \( n \to \frac{n}{2} \) (for even numbers) and \( n \to \frac{3n + 1}{2} \) (for odd numbers).
Step 1: Transformation Behavior
Case 1: \( n \) is even
When \( n \) is even: \[ n \to \frac{n}{2} \] This transformation immediately produces a smaller number because division by 2 reduces the magnitude of any positive integer \( n > 1 \).
Case 2: \( n \) is odd
When \( n \) is odd: \[ n \to \frac{3n + 1}{2} \] The result may be larger or smaller than \( n \), depending on the value of \( n \). For predictability, we aim to identify **when the sequence will drop below the initial value**.
Step 2: Predicting a Smaller Value
The sequence alternates between odd and even numbers after each transformation:
- If \( n \) is odd, apply \( n \to \frac{3n + 1}{2} \).
- If the result is even, repeatedly apply \( n \to \frac{n}{2} \) until another odd number is reached.
To predict when the sequence reaches a number smaller than \( n \), consider the "weight" of \( n \), defined by how many times it can be divided by 2 after any application of \( \frac{3n + 1}{2} \).
Key Observation
For a given number \( n \), the smallest number produced in its sequence depends on the number of times \( n \) (or subsequent terms) can be divided by 2.
Step 3: Formulating the Predictability Rule
To predict when the sequence produces a smaller number, calculate: \[ k = \text{the highest power of 2 that divides } (n-1) \] This works because:
- The transformation \( n \to \frac{3n + 1}{2} \) introduces a factor of 3 and adds 1.
- Subtracting 1 from \( n \) aligns the number with the modular behavior of powers of 2, allowing deterministic prediction of when the sequence drops below \( n \).
Example
- For \( n = 7 \):
* \( n - 1 = 6 \), and the highest power of 2 dividing 6 is \( 2^1 = 2 \). * The sequence for 7 will reach a smaller number after **1 division by 2**.
- For \( n = 15 \):
* \( n - 1 = 14 \), and the highest power of 2 dividing 14 is \( 2^1 = 2 \). * The sequence for 15 will similarly reach a smaller number after **1 division by 2**.
Step 4: Exponential Reduction
Each transformation reduces the size of the number set exponentially:
- Even numbers are solved immediately.
- Odd numbers are transformed and alternate between odd/even, with predictable reductions through divisions by 2.
- The sequence becomes smaller at a predictable rate based on the modular properties of \( n - 1 \).
Conclusion
Predictability is proven because:
- Every \( n \) eventually produces a smaller number.
- The specific step where this happens can be calculated directly from \( n - 1 \).
