Andrica’s conjecture

Conjecture (Dorin Andrica). Given the $n$th prime $p_n$, it is always the case that $1>\sqrt{p_{n+1}}-\sqrt{p_n}$.

This conjecture remains unproven as of 2007. The conjecture has been checked up to $n={10}^5$ by computers.

The largest known difference of square roots of consecutive primes happens for the small $n=4$, being approximately 0.67087347929081. The difference of the square roots of the primes ${10}^{314}-1929$ and ${10}^{314}+2318$ (which cap a prime gap of 4247 consecutive composite numbers discovered in 1992 by Baugh & O’Hara) is a very small number which is obviously much smaller than 1.