Test eq
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: <math>n \to \frac{n}{2}</math>
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: <math>n \to \frac{3n + 1}{2}</math>
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 <math>n \to \frac{3n + 1}{2}</math>.
- If the result is even, repeatedly apply <math>n \to \frac{n}{2}</math> 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 <math>\frac{3n + 1}{2}</math>.
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: <math>k = \text{the highest power of 2 that divides } (n-1)</math>
This works because:
- The transformation <math>n \to \frac{3n + 1}{2}</math> 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 <math>n = 7</math>:
* <math>n - 1 = 6</math>, and the highest power of 2 dividing 6 is <math>2^1 = 2</math>. * The sequence for 7 will reach a smaller number after **1 division by 2**.
- For <math>n = 15</math>:
* <math>n - 1 = 14</math>, and the highest power of 2 dividing 14 is <math>2^1 = 2</math>. * 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 <math>n - 1</math>.
Conclusion
Predictability is proven because:
- Every <math>n</math> eventually produces a smaller number.
- The specific step where this happens can be calculated directly from <math>n - 1</math>.
