If we were considering weather for example we could say that the weather is predictable over a period of 1 second, somewhat predictable over a period of 1 hour, and unpredictable over a period of 1 year. It doesn't matter how much historical weather data you collect you just can't say anything about the weather over long periods of time.
If a math writer says "the primes are unpredictable" this means something like if you were given a list of all the primes between 1 and some number N then there is no way to find the next prime after N that is faster than testing the primality of N+1,N+2,... until the next one is found.
The speed of the primality test doesn't matter for predictability. It doesn't matter if I can check the current weather in 1 second or 1 millionth of a second, it doesn't help me know the weather a year from now.
But not being able to work out the next prime quickly isn't the whole story. There are many things that we can say about the whole set of primes. We do know that the next prime after N won't be much bigger than N in some sense (see Bertrand's postulate). We also know roughly how many primes there are less than a given number (Prime number theorem).
There are also many properties that exist in small sets of primes (twin prime conjecture) and the OP's visualization shows some of these. So, much like the weather, the primes are not random while still being unpredictable.
If a math writer says "the primes are unpredictable" this means something like if you were given a list of all the primes between 1 and some number N then there is no way to find the next prime after N that is faster than testing the primality of N+1,N+2,... until the next one is found.
The speed of the primality test doesn't matter for predictability. It doesn't matter if I can check the current weather in 1 second or 1 millionth of a second, it doesn't help me know the weather a year from now.
But not being able to work out the next prime quickly isn't the whole story. There are many things that we can say about the whole set of primes. We do know that the next prime after N won't be much bigger than N in some sense (see Bertrand's postulate). We also know roughly how many primes there are less than a given number (Prime number theorem).
There are also many properties that exist in small sets of primes (twin prime conjecture) and the OP's visualization shows some of these. So, much like the weather, the primes are not random while still being unpredictable.