I was once teaching some middle school students math at an exclusive private school. I had one student who was arrogant and constantly annoying me and the other students. The administration was not supportive of my attempts at disciplining him. I came up with this solution:

I told him if he could find a pattern to prime numbers, so that he could predict the next one, he could make a lot of money and be famous. He liked this challenge and began devoting himself to it. He had pages and pages of calculations and never bothered me again. Every once in awhile I would show some interest in his work and he would say something like, “I think I’m on to something...”

I knew he would not find anything, because I knew that there is no pattern to prime numbers. There may be some local areas where it appears that there is a pattern, but there is no overall pattern and no formula for predicting the NEXT prime number without TESTING.

Think of it this way. You are a Paleolithic man who figures out that 2, 3, 5, 7, 11, and 13 are prime. You wonder what the next prime will be. There is no way to find it without some testing. You can test 14. Nope. 15, Nope. 16, Nope. 17, Bingo.

You only need test the factors up to the square root of the number (in the case of 17: 2, 3, and 4) because the next number will be too big, but you do need to TEST.

Mathematicians seem to hate to admit that there is this CHAOS in the middle of numbers, but there is, and I find it lovely.

Edit: How do I know that there is no pattern?

Pattern: (definition)
• an arrangement or sequence REGULARLY found in comparable objects or events.
• a REGULAR and intelligible form or sequence discernible in certain actions or situations.

So a PATTERN implies REGULARITY or REPETITION.
REPETITION implies MULTIPLICATION because MULTIPLICATION is REPETITIVE ADDITION.
Multiplication implies FACTORS, and we can’t have factors if it’s prime.

Compute: (definition) determine (the amount or number of something) mathematically.
We do not determine if a number is prime MATHEMATICALLY. We do it EXPERIMENTALLY

Edit 2: Some people think that there are “patterns” in the Ulam Spiral. Ulam spiral - Wikipedia

However, if you download the figure and blow it up you will see some straight lines emerge and then DISAPPEAR. Prime numbers are infinite. So of course statistically some straight lines will appear at times, like when flipping coins you will sometimes get a run of Heads.

Also, the Ulam Spiral uses squares. I think a different Spiral will appear if you use triangles or hexagons. It will also not produce enough of a pattern to PREDICT the next prime.

Science is about finding patterns in order to predict. We can predict when the next lunar eclipse will be, we can predict when water will freeze and boil, but we cannot predict the next prime number.

Edit 3: I think that primes don’t have a PATTERN but appear to have certain TENDENCIES. They TEND to become more SPARSE as the quantities increase, but then suddenly … you see two together. These are called twin primes. Examples: (41, 43), (137, 139). Nobody knows if twin primes, like primes, are infinite. It hasn’t been proven.

Wikipedia:
The current largest twin prime pair known is 2996863034895 · 2^1290000 ± 1
with 388,342 decimal digits. It was discovered in September 2016. Like with the primes themselves, there is no known way to predict when these twin primes will come along.

Edit 4: Steven Wolfram discusses computational irreducibility and cellular automata. Certain simple algorithms, or rules, can lead to cellular events that can not be predicted ahead of time. You have to run the program to see what will happen. The same as with prime numbers (and highly composite numbers). Computational irreducibility - Wikipedia

Edit 5: I just realized never really answered the question. There is a large prize, one million USD, to whoever can prove or disprove the Riemann hypothesis. This hypothesis involves some advanced math, but there are some good videos and books about it. It basically has important implications for the distribution of prime numbers. In other words how many primes there are below a given number. For example below 100: 25 primes; below 1000: 168 primes; below 10000: 1229 primes. According to Marcus du Sautoy, if I understand him correctly, if the hypothesis is proven to be true, that would show that primes are randomly distributed among the numbers. So no, there is no prize for anyone who discovers a pattern in prime numbers.

Summary: YOU MAY BE ABLE TO PICK UP THE SNAKE, BUT YOU DON’T KNOW WHICH WAY IT WILL TWIST NEXT.