• Collatz Conjecture

Abstract:

The Collatz Conjecture itself is fundamentally flawed. The Collatz Conjecture, as written, operates only on the coefficient of a number without regard to an unshown exponent. It therefore does not conform to mathematical principals. To correctly state the Collatz Conjecture where it adheres to mathematical principals, it should read as follows:

1. Take any number {i}x2⁰.

• If the number is odd, go to step 2.
• If the coefficient is even, normalize the coefficient:
  • Divide the coefficient by 2 and multiply the exponent by 2.
  • Repeat until the coefficient is odd.

2. When the coefficient is odd, multiply the number by 3 and add 1 to the coefficient.

3. Normalize the coefficient:

• Divide the coefficient by 2 and multiply the exponent by 2.
• Repeat until the coefficient is odd.

3. Repeat steps 2 and 3 until the coefficient normalizes to 1. Repeating step 2, the coefficient will always normalize to 1.

The Collatz Conjecture does not state the exponent function which is necessary to calculate any function of the conjecture using mathematical principals.  Each step which divides the coefficient by 2 should increment its exponent by 2 and build an exponent 2ⁿ. Every 3x+1 and normalize when applied to the coefficient will produce a value of Cx2ⁿ where C is the coefficient which has an exponent of 2ⁿ, where n>0. The conjecture, as written, has the operator toss the divide by 2 into the bit-bucket. This is mathematically unsound as the true value of the number is lost. To evaluate a coefficient without the associated exponent leads to the problems mathematicians have fought since the conjecture was introduced.

When viewed in fixed point binary, where the exponent function can be easily seen, it can be proven the least significant bit (LSB) of the coefficient functions at 4x rate (the 3x +1). The rest of the coefficient’s binary bits function only at 3x. Through multiple iterations of the Collatz process, the coefficient’s LSB continuously adds 4 to the LSB and consumes the upper bits until the true value reaches 2ⁿ where the coefficient is 1 and the exponent is 2ⁿ.