  The Firoozbakht, Nicholoson, and Farhadian conjectures can be phrased in
terms of increasingly powerful conjectured bounds on the prime gaps $g_n :=
p_{n+1}-p_n$. \[ g_n \leq p_n \left(p_n^{1/n} -1 \right)\qquad\qquad\qquad (n
\geq 1; \; Firoozbakht). \] \[ g_n \leq p_n \left((n\ln n)^{1/n} -1
\right)\qquad\qquad (n>4; \; Nicholson). \] \[ g_n \leq p_n \left( \left(p_n
{\ln n\over\ln p_n}\right)^{1/n} -1 \right)\qquad (n>4; \; Farhadian). \] While
a general proof of any of these conjectures is far out of reach I shall show
that all three of these conjectures are unconditionally and explicitly verified
for all primes below the location of the 81$^{st}$ maximal prime gap, certainly
for all primes $p <2^{64}$. For the Firoozbakht conjecture this is a very minor
improvement on currently known results, for the Nicholson and Farhadian
conjectures this may be more interesting.
