Last week, I made a mention to a fun prime number theorem. That theorem is the Green Tao theorem. I’ve been trying to get enough information on it so that I could really put together a comprehensive review of the theorem… but I tried getting through the actual write up… and frankly it was a bit too dry for me.
On the surface, I suppose the theorem is reasonably simple. It states that for any number k there exist infinitely many arithmetic progressions of length k consisting of prime numbers.
Let’s look at some basic prime numbers.
2, 3, 5, 7, 11, 13, 17, 19
in this series, numbers 3, 5, and 7, are two numbers apart. 7-5 = 2 and 5-3 = 2.
Since there are three numbers in this series, k = 3
Likewise 7, 13, and 19 are equally spaced. With 6 integers between them. 19-13 = 6, 13-7 = 6
This series also has length k = 3
Ben Green and Terrence Tao (Mathematician Rockstars) are suggesting that you can find strings like this of ANY length.
So, if I took a number, say… a billion = k, this theorem states that yes there are a billion primes out there that have equal arithmetic spacing. That disgusts me with its insanity.
That means if you took an infinite number line, you could take a comb with any number of equally spaced teeth, and scale the comb up or down until you were able to place it on the number line so that every tooth aligned with a prime number. That blows my mind.