Logitech C920 - Not Recognized by Windows 11

For me, the issue was caused by the USB ports. The webcam would not work on the USB hub that I was using with my laptop, a ThinkPad, and it would only work on certain ports on the ThinkPad USB-C Dock Gen 2.

Windows 10 Polish Key Combinations

Install the Polish language pack. Then use the following key combinations for the Polish characters:

Right-Alt + Z = ż
Right-Alt + X = ź
Right-Alt + S = ś
Right-Alt + C = ć
Right-Alt + L = ł
Right-Alt + O = ó
Right-Alt + E = ę
Right-Alt + A = ą
Right-Alt + N = ń

A Simple Collatz Proof

A Simple Collatz Proof Proof by exhaustion and reductio ad absurdum.

1. It has already been proven that the Collatz conjecture is true for a large set of odd numbers. This is the set Po. The largest number in the set is Pn.
2. The Collatz series for all numbers n repeats at the interval 4n + 1. This set, the extended set of the Collatz series for all odd numbers, is the set Pi. For Pn, there are an infinite subset of numbers in Pi that are greater than Pn. This is the set, Pm.
3. Consider the next highest odd number, an odd number that is the largest number in the Po set plus two: N = Pn + 2.
4. Either some number in the Collatz series for N will be equal to a number in the Pm set or a number in the Po set, or they will not. If no number in the Collatz series for N is equal to a number in either set, there would exist an infinite set of odd numbers where the Collatz conjecture is false, with a series of numbers that never intersect with the series of numbers where the conjecture has been proven true. Given the size of the Pm set and the Po set, this is impossible! Therefore, the conjecture is true for all numbers.
5. Q.E.D.

Note: I separated out the Pm set to show that for any odd number that is tested, there will always be an infinite set of odd numbers known to be true for Collatz that are greater than the number being tested. So, even when the numbers in a series are rising, they can still intersect with a number that, indirectly, has been proven true for the conjecture.

Example for step 2: For n = 27, the Collatz series will be the same for 27, 4(27) + 1, 4(4(27) +1) + 1, 4(4(4(27) +1) + 1) + 1, and so on. And this would extend with all of the numbers that make up the Collatz series for 27: 41, 31, 47, 71, 107, 161, 121, 91, 137, 103, 155, 233, 175, 263, 395, 593, 445, 167, 251, 377, 283, 425, 319, 479, 719, 1079, 1619, 2429, 911, 1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, and 1.