  A strong version of Andrica's conjecture can be formulated as follows: Except
for $p_n\in\{3,7,13,23,31,113\}$, that is $n\in\{2,4,6,9,11,30\}$, one
has$\sqrt{p_{n+1}}-\sqrt{p_n} < \frac{1}{2}.$ While a proof is far out of reach
I shall show that this strong version of Andrica's conjecture is
unconditionally and explicitly verified for all primes below the location of
the 81$^{st}$ maximal prime gap, certainly for all primes $p <2^{64}\approx
1.844\times 10^{19}$. Furthermore this strong Andrica conjecture is slightly
stronger than Oppermann's conjecture --- which in turn is slightly stronger
than both the strong and standard Legendre conjectures, and the strong and
standard Brocard conjectures. Thus the Oppermann conjecture, and strong and
standard Legendre conjectures, are all unconditionally and explicitly verified
for all primes $p <2^{64}\approx1.844\times 10^{19}$. Similarly, the strong and
standard Brocard conjectures are unconditionally and explicitly verified for
all primes $p <2^{32} \approx 4.294 \times 10^9$.
