Root prime gap
by John from John D. Cook on (#74TAT)
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 pn and pn+1 are consecutive prime numbers, then
pn+1 - pn < 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:
pn+1 < 1 + 2pn + pn,
which would be an improvement on the Bertrand-Chebyshev theorem that says
pn+1 < 2pn.
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