# 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.

## References

• 1 D. Baugh & F. O’Hara, “Large Prime Gaps” J. Recr. Math. 24 (1992): 186 - 187.
Title Andrica’s conjecture AndricasConjecture 2013-03-22 16:41:22 2013-03-22 16:41:22 PrimeFan (13766) PrimeFan (13766) 8 PrimeFan (13766) Conjecture msc 11A41 Andrica conjecture