Simon Plouffe has found a formula which produces 50 prime numbers in a row:

\[\begin{align}

a_0 &= 10^{500}+961 \\

a_{n+1} &= a_n^{\frac{101}{100}}

\end{align}\]

\(\lfloor a_n \rfloor\) is a prime for the first 50 \(a_n\)!

He's trying to find better formulas than those discovered by Wright and Mills, which produce infinitely many primes but grow really quickly.

Paper on the arXiv: https://arxiv.org/abs/1901.01849

(thanks @erou)