- Welcome to the Explanation of Collatz Conjecture
By
Terrence Mummert
The study of Collatz Conjecture using Floating Point Binary Mathematics
Abstract:
The Collatz Conjecture itself is fundamentally flawed. The Collatz Conjecture, as written, is a process but not an equation. The conjecture operates on the operand of a number without regard to the characteristic. By simply discarding the divide-by-2, the rules of mathematics are also discarded. Only when the divide-by-2 is considered within a full equation for Collatz can the riddle be solved.
By applying the principles of Scientific Notation to the Collatz Conjecture, it then conforms to mathematical functions and principals.
- Define the Collatz Process as the traditional step by step execution and is referred to in this paper as the Collatz Value
- Define the Collatz Equation as the (Collatz Process) * 2n.
- The Collatz Equation value will be referred to in this paper as the True Value.
- The True Value is anchored to a fixed binary point.
- As each divide-by-2 occurs, the characteristic of 2n increments, left one bit position for each divide-by-2.
Traditional views of the Collatz Conjecture have the divide-by-2 applied to the operand with no corresponding equation multiplier. In the below, the binary 5 will shift right for each divide-by-2 and the trailing zeros shift out of existence.
The true value below is 1280. Both the operand and characteristic are easily identifiable. The next 3x and plus are applied to the operand. The true plus 1 is (1 x the current characteristic of 2n).
By maintaining the integrity of the True Value, the Collatz Equation now follows all the rules of mathematics. The 3x step and the plus 1 step increase the operand. The divide-by-2 steps increase the characteristic.
- This paper proves the Collatz Conjecture is fundamentally flawed by not recognizing the characteristic.
- This paper proves, when viewed in binary, that as the Collatz Equation proceeds, the operand is multiplied by 3. So, the operand is increasing at a 3x rate, bit for bit. However, the least significant bit of the operand, being solely impacted by the plus 1, increases at a 4x rate. In so doing, the lower bits overtake the upper bits until the True Value is 2n. This is proof the Collatz Process is correct.
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The Collatz Unwound
The Collatz Conjecture is, at best, misleading. The Collatz conjecture as written is a process, but not an equation. As a process, using mathematical principals to solve leads to impossibilities. The truth, Collatz never reaches 1. But rather, it achieves a value of 1 * 2n, where n>0. And further, the true value of any step in the Collatz conjecture, after the entrance of the initial odd value, will always be an even number due to the characteristic of 2n.
Math demands an equation and the Collatz Conjecture is just a process. A fragment of a true equation. This process simply discards the divide by 2. But to become an equation, each divide-by-2 step is applied to a 2n multiplier. Consider the Collatz equation as (The Collatz Process) x 2n. Each divide-by-2 step clocks n+1.
The Collatz conjecture operates best in the world of Floating Point Binary. And, as written, the conjecture operates only the mantissa of an equation with no regard for the characteristic. With each divide by 2 cycle, Collatz simply discards each division of 2. That is fine for a process, but it is not mathematically sound. To truly evaluate Collatz, the divide by 2 must be accounted for.
In the below case for 15, Collatz conjecture is put into an equation which includes the multiplier, 2n. With 2n in the equation and viewed in binary, everything needed to understand the Collatz Conjecture is demonstrated in the below case for 15 and can be applied for any {i}. Below, the Collatz value column contains the values derived from the traditional execution of the Collatz process. The second column contains the true values derived using the full equation, (The Collatz Process) x 2n. The traditional Collatz process values can easily be masked from the full true value as demonstrated below. The characteristic of 2n is also easily masked from the true value. The key to Collatz is not shifting the value toward the binary point. To see the Collatz Process value, shift the binary point left. This also reveals the true value at each step.
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The case for 15 demonstrates the where the characteristic of 2n is needed. The Collatz Process are easily tracked in relation to their characteristic.
The above demonstrates:
- The Collatz Conjecture as writen is only a fragment of an equation.
- Divides must be accounted for in math.
- A Collatz iteration never reach 1. Rather 1 x 2n.
- The Collatz conjecture operates only in the mantissa of a floating-point number without regard for the characteristic of 2n.
Each step in solving Collatz for the initial value of 15 is shown above. It can be seen where true value of a complete iteration never reaches 1. But rather, it reaches 2n where n>0. Looking after the first divide by 2, the true value has a characteristic of 21. Looking at the final divide by 2, the true value arrives at 1x 212.
Collatz, when he defined the conjecture, omitted a vital part of math, the characteristic. To be mathematically correct, for each divide by 2 step, the characteristic must be multiplied by 2 for Collatz to be a true equation.
Note, there are 2 basic components represented in each step, the mantissa and the characteristic. The Collatz Conjecture process only operates on the mantissa. Note how the left mask exactly matches the Collatz values for 15. The righthand mask is the characteristic. Collatz does not enumerate the characteristic in his conjecture. As such, it has been overlooked until now. Each divide by 2 must be accounted for.
While the plus 1 in the second iteration does add a Collatz 1, the process is multiplied by the characteristic showing the true value addend is 2. This applies to the Collatz process through each iteration. Note how the 2n increases in value for each divide by 2 in the iteration for the Collatz value of 159. Note how the 5 divide by 2 step clock the 2n from 23 to 28. So, the next step begins with a Collatz 5 but has a true value of 1280 or 5 x 28. And the true value of the addend for the iteration of 5 is 256, or 1 x 28. The mantissa is the missing part of the Collatz process as defined by his conjecture. However, the conjecture is not an equation. The 2n must be accounted for to make the Collatz conjecture into an equation.
This process is valid for any starting value. 15 was used as it fits best in the webpage.
Please read on.
The Collatz Conjecture Unwound
The Collatz Conjecture is, at best, misleading. The Collatz conjecture as written is a process, but not an equation. As a process, using mathematical principals to solve leads to impossibilities. The truth is Collatz never truly reaches 1, but rather achieves a value of 1 * 2n, where n>0. And further, the true value of any step in the Collatz conjecture, after the entrance of the initial odd value, will always be an even number due to the characteristic of 2n.
The Collatz conjecture operates best in the world of Floating Point Binary. And, as written, the conjecture operates only the mantissa of an equation with no regard for the characteristic. With each divide by 2 cycle, Collatz simply discards each division of 2. That is fine for a process, but it is not mathematically sound. To truly evaluate Collatz, the divide by 2 must be accounted for.
The true equation:
[Collatz process]*2n, where the 2n clocks up plus 1 for each Collatz process divide by 2.
A simple inspection where x=1, 3*1+1=4. 4=1*22 demonstrates this clearly. The next iteration, according to the Collatz process, the 3x will only be applied to the mantissa of 1. 1*3+1=4. And 4 divides by 2 twice to equal 1. Where 3*4+1=13 or 3*(1*22)=13, Collatz will have us to believe it again returns to 1, as Collatz only applies the multiple of 3 to the mantissa and the characteristic of the number is not also multiplied by 3 and simply discarded. Here, Collatz is proven to not be an equation at all.
Collatz is a binary process. The divide by 2 function mandates inspection in a binary format. Because the missing characteristic is a value of 2n, the visualization of the divide by 2 will appear as trailing zeros a the Collatz iterations progress. An actual divide by 2 will move the radix point to the left for each Collatz iteration.
In the Case for x=15 illustration, the characteristic (the part the Collatz conjecture leaves out) can be extracted by dividing the binary representation by 2, as Collatz prescribes by simply deleting all the trailing zeros. But again, Collatz conjecture is a method but not an equation and these zeros must be accounted for to render a mathematical solution.
Normally, in Collatz, the values are right shifted for each divide by 2. Doing so, the parity between the Collatz value and the True Value would be lost. The below chart shows the Collatz process with x=15 using the properties of floating point binary representation. In iteration 1, 3×15=45. 45+1=46 46/2=23. This is what Collatz would have you believe. The reality is 46/2 can be rewritten as 23*21. 46/2 = 23*21. This is the trick of Collatz. Collatz will have the subsequent 3x apply only to the mantissa of 23, with no regard for the characteristic of 21. In iteration 2, the Collatz process ceases to be an equation. However, in the True Value column, the equation applies the characteristic to both the mantissa and to the addend 1. 3*23*21 + 1*21. Here is where a binary representation truly stands out. Not there is no variant between the bit structure of 23 and 46. Only the placement of the binary point changes. In this case, the binary point for the Collatz representation is green where the True Value binary point is magenta. When the Collatz process applies the “Divide by 2(s)”, the binary point simply shifts left for each case until the number goes odd. All odd numbers in binary end in 1 where even numbers end in 0. Simply shift the binary point left pass all trailing 0’s to reveal the Collatz value. But the trailing o’s must be accounted for.
Note: The 2x value is shown in the below example. Please refer to the 1x+2x=3x link on the home page for the complete explanation as to why that is relevant.
Case for x=15
(Note: the use of the 2x value will be explained in a later section. It is an important part of the Collatz process. Iteration 5 will provide some clues to this.)
The trick to Collatz lays in the True Value of the Collatz “Plus 1”. Because the Collatz process operates the multiply by 3 only on the mantissa, the “Plus 1” is also referencing only the mantissa. Where Collatz will have the belief where only 1 is the addend, the true addend for each iteration is:
Collatz Value | True Value | |
Iteration 1 | 1 | 1 |
Iteration 2 | 1 | 2 |
Iteration 3 | 1 | 4 |
Iteration 4 | 1 | 8 |
Iteration 5 | 1 | 256 |
The Collatz Conjecture Unwound
Collatz conjecture is at best, misleading. The Collatz conjecture as written is a process, but not an equation. As a process, using mathematical principals to solve leads to impossibilities. The truth is Collatz never truly reaches 1, but rather achieves a value of 1 * 2n, where n>0. And further, the true value of any step in the Collatz conjecture, after the entrance of the initial odd value, will always be an even number due to the characteristic of 2n.
The Collatz conjecture operates best in the world of Floating Point Binary. And, as written, the conjecture operates only the mantissa of an equation with no regard for the characteristic. With each divide by 2 cycle, Collatz simply discards each division of 2. That is fine for a process, but it is not mathematically sound. To truly evaluate Collatz, the divide by 2 must be accounted for.
The true equation:
[Collatz process]*2n, where the 2n clocks up plus 1 for each Collatz process divide by 2.
A simple inspection where x=1, 3*1+1=4. 4=1*22 demonstrates this clearly. The next iteration, according to the Collatz process, the 3x will only be applied to the mantissa of 1. 1*3+1=4. And 4 divides by 2 twice to equal 1. Where 3*4+1=13 or 3*(1*22)=13, Collatz will have us to believe it again returns to 1, as Collatz only applies the multiple of 3 to the mantissa and the characteristic of the number is not also multiplied by 3 and simply discarded. Here, Collatz is proven to not be an equation at all.
Collatz is a binary process. The divide by 2 function mandates inspection in a binary format. Because the missing characteristic is a value of 2n, the visualization of the divide by 2 will appear as trailing zeros a the Collatz iterations progress. An actual divide by 2 will move the radix point to the left for each Collatz iteration.
In the Case for x=15 illustration, the characteristic (the part the Collatz conjecture leaves out) can be extracted by dividing the binary representation by 2, as Collatz prescribes by simply deleting all the trailing zeros. But again, Collatz conjecture is a method but not an equation and these zeros must be accounted for to render a mathematical solution.
Normally, in Collatz, the values are right shifted for each divide by 2. Doing so, the parity between the Collatz value and the True Value would be lost. The below chart shows the Collatz process with x=15 using the properties of floating point binary representation. In iteration 1, 3×15=45. 45+1=46 46/2=23. This is what Collatz would have you believe. The reality is 46/2 can be rewritten as 23*21. 46/2 = 23*21. This is the trick of Collatz. Collatz will have the subsequent 3x apply only to the mantissa of 23, with no regard for the characteristic of 21. In iteration 2, the Collatz process ceases to be an equation. However, in the True Value column, the equation applies the characteristic to both the mantissa and to the addend 1. 3*23*21 + 1*21. Here is where a binary representation truly stands out. Not there is no variant between the bit structure of 23 and 46. Only the placement of the binary point changes. In this case, the binary point for the Collatz representation is green where the True Value binary point is magenta. When the Collatz process applies the “Divide by 2(s)”, the binary point simply shifts left for each case until the number goes odd. All odd numbers in binary end in 1 where even numbers end in 0. Simply shift the binary point left pass all trailing 0’s to reveal the Collatz value. But the trailing o’s must be accounted for.
Note: The 2x value is shown in the below example. Please refer to the 1x+2x=3x link on the home page for the complete explanation as to why that is relevant.
Case for x=15
(Note: the use of the 2x value will be explained in a later section. It is an important part of the Collatz process. Iteration 5 will provide some clues to this.)
The trick to Collatz lays in the True Value of the Collatz “Plus 1”. Because the Collatz process operates the multiply by 3 only on the mantissa, the “Plus 1” is also referencing only the mantissa. Where Collatz will have the belief where only 1 is the addend, the true addend for each iteration is:
Collatz Value | True Value | |
Iteration 1 | 1 | 1 |
Iteration 2 | 1 | 2 |
Iteration 3 | 1 | 4 |
Iteration 4 | 1 | 8 |
Iteration 5 | 1 | 256 |