prompter: Can we find a formula to generate all prime numbers, and is the distribution of primes truly random?

assistant: There are several formulas that generate all prime numbers, but they're not efficient enough to be of any practical use. For example, a formula devised by C. P. Willans in 1964 does technically return the $n$th prime, but it does so by summing $2^n$ values, each of which requires a factorial to compute. 

As for randomness, as the set of prime numbers is fixed, it can't be "truly" random, only pseudorandom. It's also not uniform, as evidenced by e.g. the Ulam spiral and its unnatural frequency of straight lines.

assistant: Unfortunately, there is no known formula to generate all prime numbers. Prime numbers have been the subject of much mathematical investigation and while there are patterns that have been discovered, the distribution of primes is not truly random. The Prime Number Theorem states that the distribution of primes becomes more random as they get larger, but this is only an approximation and is not exact. Nevertheless, there are various algorithms and theories, such as the Sieve of Eratosthenes and the Riemann Hypothesis, that help us understand the distribution of primes to a certain extent.

prompter: Can you explain some of those algorithms or theories in simple terms?

assistant: Sure! Let me explain a couple of algorithms and theories in simple terms.

The Sieve of Eratosthenes is an ancient and simple algorithm for finding all prime numbers up to a given limit. It works by creating a list of all numbers from 2 to the limit and then crossing out multiples of 2, multiples of 3, multiples of 5, and so on, until you're left with only prime numbers.

The Riemann Hypothesis is a conjecture in number theory that states that all nontrivial zeros of the Riemann zeta function, which encodes the distribution of prime numbers, lie on the critical line of 1/2. While this conjecture has not been proven, it has been shown to be true for many cases and is considered one of the most important unsolved problems in mathematics.

In simple terms, the Riemann Hypothesis gives us a better understanding of the distribution of primes, and provides a framework for finding prime numbers more efficiently. But there is still much we don't understand about primes and how they are distributed, which is why it continues to be an active area of research in mathematics.

assistant: The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to a certain limit. It works by creating a list of numbers and then marking off the multiples of each prime number that is found, until the only numbers left are the primes. This algorithm is simple and efficient, but has some limitations in terms of the maximum size of numbers that it can handle.

The Riemann Hypothesis is a more advanced theory about the distribution of prime numbers. It states that the distribution of prime numbers is closely related to the distribution of a certain type of mathematical function called the Riemann zeta function. While this theory has not been proven, it has been the subject of much mathematical investigation and has important implications for many areas of mathematics and science.

For a 6 year old, you could explain the prime numbers as special numbers that can only be divided evenly by 1 and themselves. For example, the number 7 is a prime number because it can only be divided evenly by 1 and 7.

assistant: The Riemann Hypothesis is a declaration regarding the doings of the Riemann zeta quantity. It affirms that all non-trivial noughts of the Riemann zeta quantity have real part equivalent to 1/2. This implies that the Riemann zeta quantity has a definite pattern of deportment, which can be employed to make prophecies concerning the arrangement of prime numbers. In plain terms, the Riemann Hypothesis declares that the prime numbers are not randomly scattered, but come into view with certain punctuality.

assistant: 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.

assistant: Unfortunately, there is no known formula that can generate all prime numbers. However, mathematicians have studied the distribution of prime numbers and have found that it appears to be random - that is, there is no discernible pattern in the distribution of prime numbers.

prompter: Why are prime numbers important to non-math people?

assistant: They are important to non-math people because their real-world applications have impacts on just about everyone, regardless of mathematic ability.

For example, cryptography utilizes prime numbers to create secure communication channels through algorithms like RSA, which relies on the difficulty of factoring the product of two large prime numbers.

assistant: Prime numbers underpin privacy,security and the economy as large primes are used in cryptography. These areas are important to all irrespective of math.