Someone on here asked a question or mentioned something recently.  I think the discussion was about "wrong" ideas that are useful, and the idea was that prime numbers are random.  Someone in replied that prime numbers are not random.  But that got me thinking, do random numbers exist?  I have an engineering and computer science background.  There are lots of methods to produce "random numbers" and "random distributions".

But that really got me thinking, is it even possible to actually randomly choose a number?  I can't think of a method or model that is truly random.  That is to say, with the same set of inputs (seed) you will always get the same number out.  Even if you asked a human to pick a random number out of a hat, most of them have some cognitive bias.  You might also argue that if you went back in time, and asked them to pick a number, the same neurons would fire they would pick the same number.

So if it is argued that prime numbers are not random, because they are determined by some deterministic rule/formula, what IS an example of a random number?

If you argue that say you choose a random number from a distribution, HOW could it be possible to randomly choose that number?
Radioactive decay (and quantum tunneling in general) seems to be genuinely random.
You are right that you can't use a deterministic process to choose random numbers, so you will have to use a random physical process.

Radioactive decay and thermal noise are two well regarded mechanisms.  Thermal noise has good properties, great entropy, and can be a good source.

There are mechanisms to remove bias from these sources (pink noise vs white noise, etc).

Generally, these random bits are fed into strong cipher algorithms to act as a reservoir of entropy that can be pulled as needed.
There is a method for detecting fluctuations in the atmosphere's thickness that has been applied to adaptive optics on ground based telescopes. I am pretty sure that the fluctuations are random and have been used as a RNG.
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How do you define random process?(not Stochastic process).
For me, random process is the process with no condition on the process.
Prime number are not random as we define it with condition that it's not a composite number.
>So if it is argued that prime numbers are not random, because they are determined by some deterministic rule/formula, what IS an example of a random number?

Isn't one of the big open questions if there is such a rule/formula? As far as I know there is no function which will produce you the n-th prime number, nor is there even a function which will produce an endless stream of unique prime numbers. Thus, while following a certain statistical distribution, I would say prime numbers are random.
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Brownian Motion
In computing, particularly in cryptography, the sense of “random” that we really care about is “unpredictable”. It doesn’t matter that a PRNG or hash is deterministic, it matters that an adversary has no feasible way to predict the next output, nor derive the input. A true uniform random number generator may or may not correspond to a real physical phenomenon, but that’s a question for a physicist—we can still define a good PRNG by how hard it is to tell apart from that idealised model. In other words, it’s always _possible_ to predict the output of a generator or break a hash, we only need it to be _improbable_, for example, requiring brute-force search over a very large space.
I believe the overwhelming majority of real numbers would have genuinely random (as in not predicted by any algorithm) decimal representations as a consequence of the fact that the set of possible  algorithms is only countably infinite while the set of real numbers is uncountably infinite. Whether or not such numbers really exist in any sense is as debatable as whether any numbers really exist, but they are called *real* numbers.
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True random numbers is an ideal with no definitive test for it. There are thousands of randomness tests. But passing all of them doesn’t mean a set of numbers is truly random.
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