Collatz conjecture: Difference between revisions
(Replaced content with "== 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...") Tag: Replaced |
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## 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). | 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: | When \( n \) is even: | ||
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This transformation immediately produces a smaller number because division by 2 reduces the magnitude of any positive integer \( n > 1 \). | 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: | When \( n \) is odd: | ||
\[ | \[ | ||
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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**. | 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: | The sequence alternates between odd and even numbers after each transformation: | ||
1. If \( n \) is odd, apply \( n \to \frac{3n + 1}{2} \). | |||
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 \( \frac{3n + 1}{2} \). | 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. | 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: | To predict when the sequence produces a smaller number, calculate: | ||
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\] | \] | ||
This works because: | This works because: | ||
1. The transformation \( n \to \frac{3n + 1}{2} \) introduces a factor of 3 and adds 1. | |||
2. 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 \( n = 7 \): | * For \( n = 7 \): | ||
* \( n - 1 = 6 \), and the highest power of 2 dividing 6 is \( 2^1 = 2 \). | * \( n - 1 = 6 \), and the highest power of 2 dividing 6 is \( 2^1 = 2 \). | ||
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* The sequence for 15 will similarly reach a smaller number after **1 division by 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: | Each transformation reduces the size of the number set exponentially: | ||
1. Even numbers are solved immediately. | |||
2. Odd numbers are transformed and alternate between odd/even, with predictable reductions through divisions by 2. | |||
3. The sequence becomes smaller at a predictable rate based on the modular properties of \( n - 1 \). | |||
### Conclusion | |||
Predictability is proven because: | Predictability is proven because: | ||
1. Every \( n \) eventually produces a smaller number. | |||
2. The specific step where this happens can be calculated directly from \( n - 1 \). | |||
Revision as of 09:19, 20 December 2024
- 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: 1. If \( n \) is odd, apply \( n \to \frac{3n + 1}{2} \). 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: 1. The transformation \( n \to \frac{3n + 1}{2} \) introduces a factor of 3 and adds 1. 2. 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: 1. Even numbers are solved immediately. 2. Odd numbers are transformed and alternate between odd/even, with predictable reductions through divisions by 2. 3. The sequence becomes smaller at a predictable rate based on the modular properties of \( n - 1 \).
- Conclusion
Predictability is proven because: 1. Every \( n \) eventually produces a smaller number. 2. The specific step where this happens can be calculated directly from \( n - 1 \).
