Blog post: Unity+: Connection between the function e^x and the Collatz conjecture

The function of e^x, or simply the constant of e, has many special properties that makes it a unique function. For example, the derivative of e^x is simply itself as its own derivative. This has lead to one particular discovery about the connection between the function e^x and the Collatz conjecture, which is the following.

This is what is called a Lambda Function of the Aleph Derivative. This takes two functions, like in the Collatz conjecture, and makes an equation to describe the trend of the output of the rules, such as found in the Collatz conjecture. More specifically, it takes the two concepts, which are the Aleph Integral and the derivative of the function and its inverse, and combines them together to form a function to describe the trend of the Collatzian ratio and the effect from using the rules found in the Collatz conjecture. For example, here is what describes the parameters of the Collatz conjecture and the function of the trends.

This means that the function that describes the trend of growth for the parameters of the Collatz conjecture is e^(x/2). This means the trend will grow exponentially in terms with the natural constant e.

The mean, or average, of all similar trends will eventually become the function of e^x. This means that all values of n for the Collatz parameters will end up producing a finitely sized Hailstone sequence. This means that the Collatz conjecture must be true because it will remain on a finite portion of the function of e^x.

Then the question becomes what will happen when dealing with Collatzian ratios that becomes polynomial equations or involve sine or cosine functions. For example, here is a graph that would represent the output of these kind of functions in a Collatzian ratio.

Based on the conclusion of the exponential function, the conclusion that can be made from this type of function is that it would be a looping sequence. This means that it would never approach 1. There are also other type of functions that would reach this conclusion, but this is a very simple example of this.

I will continue research on this. Thanks for reading.
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