Find all prime $a, b, c$ such that $ab+bc+ac > abc$ Find all primes $a, b, c$ such that $ab + bc + ac > abc$

let $a\geq b \geq c$
if $c\geq 3$ then $ab+bc+ac\leq 3ab\leq cab\implies c=2$.
We now need $ab+2(a+b)> 2ab\iff 2(a+b)>ab$
if $b\geq 4$ then $2(a+b)\leq4a \leq ab$, so $b\in \{2,3\}$
if $b=2$ then $2(a+2)>2a$.
if $b=3$ then $2a+6>3a\iff a<6$.
Hence ordered solutions are $(p,2,2)$ and $(3,3,2),(5,3,2)$
