Diophantus Era begins!

The key fact is that $x^5+x^4+1 = (x^3-x+1)(x^2+x+1).$ Also note that $(x^3-x+1)(x+3) - (x^2+x+1)(x^2+2x-4) = 7.$ (This comes from a standard Euclidean algorithm computation.)

So if the two factors' product is a power of some prime $p,$ then they're both powers of $p$ as well, so their gcd equals the minimum of the two of them. But the second equation above shows that that gcd divides $7.$

So either one of the factors is $1,$ or one of the factors is $7.$ The first case has one positive solution for $x,$ namely $x=1,p=3,y=1.$ For the second case, the only positive solution to $x^2+x+1=7$ and the only positive solution to $x^3-x+1=7$ is $x=2$ and $x=2.$ Indeed, $x=2,p=7,y=2$ is a solution. These are the only cases, so we're done, and the answer is $\fbox{2}.$