<s>[INST] Can we find a formula to generate all prime numbers, and is the distribution of primes truly random? [/INST] As of right now, there is no known simple formula that generates prime numbers. I am not aware of any mathematical proofs regarding whether such a formula must exist or cannot exist either. However, there are algorithms that can determine whether a number is prime. For example, one can calculate the modulo of all smaller prime numbers up to the square root of the number to be tested; if the modulo with any of those smaller numbers is equal to zero, then the number to be tested is a composite number, otherwise it is a prime number. Other, more complex algorithms exist to for example calculate Mersenne primes, which are primes that precede a power of two.

As for whether the distribution of primes is "truly random": this depends on which definition of probability you are using. Let us consider the frequentist definition of probability first. It is defined as the fraction of random experiments that satisfy a condition if an infinite number of experiments were to be conducted. Using this definition the distribution of primes is not random since a number being prime is a fixed property that does not change upon repetition. Let us now consider Bayesian probability: it is defined as the subjective confidence in a hypothesis being true. For a number whose primality has not yet been tested the primality can then be assumed to be random in the sense that while the number definitely is either prime or composite, we cannot yet say with certainty which is the case. The Bayesian probability of an unknown number being prime is proportional to the number of its digit, i.e. its logarithm. I am not aware of any pattern that emerges from the distribution of prime numbers that could be distinguished from randomness.</s>