  Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap.
We demonstrate the existence of infinitely many values of $n$ for which
$g_n>g_{n+1}>\cdots>g_{n+m}$ with $m\gg \log\log\log n$ and similarly for the
reversed inequalities. In doing so we settle a conjecture of Erd\"os for the
case $m=2$.
