Root prime gap

I recently found out about Andrica’s conjecture: the square roots of consecutive primes are less than 1 apart.

In symbols, Andrica’s conjecture says that if p_n and p_n+1 are consecutive prime numbers, then

√p_n+1 − √p_n < 1.

This has been empirically verified for primes up to 2 × 10¹⁹.

If the conjecture is true, it puts an upper bound on how long you’d have to search to find the next prime:

p_n+1 < 1 + 2√p_n  + p_n,

which would be an improvement on the Bertrand-Chebyshev theorem that says

p_n+1 < 2p_n.