Is there a formula to predict the next prime number?

**Short answer:** No polynomial formula can generate only primes (except constant functions). **Proof:** If $f(n)$ is a non-constant polynomial with integer coefficients that generates primes for all integer inputs $n$, then $f(0) = p$ for some prime $p$. Then $f(kp) \equiv f(0) \equiv p \pmod{p}$, so $f(kp)$ is divisible by $p$. Since $f(kp) \to \infty$, eventually $f(kp) > p$, so it would be a larger multiple of $p$ and thus composite. Contradiction. **What about exponential formulas?** The formula $2^{2^n} + 1$ (Fermat numbers) gives primes for $n=0,\ldots,4$ but fails at $n=5$ (composite). Mersenne primes $2^p-1$ are prime for some $p$ but not all. **Whats known:** - There are formulas that generate primes, but theyre either computationally useless or require knowing primes in advance - Millss constant: There exists $\theta > 0$ such that $\lfloor \theta^{3^n} \rfloor$ is always prime — but finding $\theta$ requires knowing primes - The distribution of primes follows $\pi(x) \sim x/\ln x$ but exact prediction is impossible **Why its hard:** Primes are determined by multiplicative structure, but our formulas are built on addition and exponentiation. These two operations interact in complex ways.