Definitions

The Syracuse Conjecture is certainly one of the few open mathematical problems whose statement is understandable by all, that is why it is so fascinating.

I call a revolutionary number, in the sense of the Syracuse sequence, a number such that at the end of a certain number of steps in the Syracuse sequence (standard or reduced), has again a value very close to the starting number.
So after a very chaotic journey, we almost did a cycle, which is like a revolution.
As a reminder, if there was a non-trivial cycle, different from (4,2,1), then the Syracuse conjecture would be false, which gives mathematical interest to these numbers.
For purists, it is a number such that there exists N for which the relative difference between u(N) and u(0) is very small, which means that we have "almost" a cycle for u(0), let |u(N) / u(0) - 1| < ε, ε being small. u representing the standard Syracuse sequence, u(n+1) = u(n)/2 if u(n) is even, and u(n+1) = 3×u(n)+1 if u(n) is odd.

We will call v, the reduced Syracuse sequence, defined as follows: v(n+1) = v(n)/2 if v(n) is even, and v(n+1) = (3×u(n)+1)/2 if v(n) is odd.

Example

It's a revolutionary number with ε = 4.366×10^-5 for the Syracuse sequence, so with a relative difference of a few hundred thousandths.
It also has a signature including in particular a serial number (SN) which is equal to 0.
With all of these properties, there is theoretically less than a 1 in 10¹⁰⁰ chance of finding one randomly, which means that it would be easier to mine all the remaining bitcoins.
This fits modestly into a much larger mathematical problem: Easy to check, easy to find? Or easy to verify, difficult to forge
The jewel is here: Example of revolutionary number and digital ticket based on the result of Syracuse
To check, click the "Compute" button and look at the result in the "List of steps written as string of ASCII characters" text box, where SN represents the serial number of the ticket.
You will also have all the details on the precision, as well as the plot of the trajectory (be careful, the scale is logarithmic) even until it reaches the value 1 after 6957 steps.
You can see the results in this PDF file: jb_syr_sn_0_en.pdf

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These numbers have the same properties as the one in the example, their Serial Number (SN) is greater than 1,000,000.
The lower the SN serial number, the more searched it will be.
Keep your numbers in a safe place, you will be able to marvel at the result, everything is vibration including these numbers ... it's magic!
Whether you are passionate about the Syracuse conjecture, collector or speculator, thank you in advance for showing me a little sympathy

Verification

To check a number, you can program the Syracuse sequence, it is very simple for small numbers, a little more complicated for "large" numbers (greater than 128 bits) but there are public libraries for these calculations.
You will find here an implementation in Javascript which has the advantage of working in an Internet browser: https://www.bajaxe.com/jb421/jb421_verif.html?lg=en&type=red
Check that the type of Syracuse sequence is reduced, which should be the case with the link
You just have to specify your starting number, the number of desired transitions and click on the "Compute" button
You will also have all the details on the precision, as well as the plot of the trajectory (be careful, the scale is logarithmic) even until it reaches the value 1 if you check the option.