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 → n/2 (for even numbers) and n → (3n + 1)/2 (for odd numbers).

Step 1: Transformation Behavior

Case 1: n is even When n is even: n → 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 → (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 → (3n + 1)/2.
• If the result is even, repeatedly apply n → 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 (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 = the highest power of 2 that divides (n-1) This works because:

• The transformation n → (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.