Context-free string that satisfies constraint?
It is not necessary to define a string that has exactly length $p$. It only has to have length at least $p$. So $(i,j,k) = (2p^2,2p^2,2p)$ so that $2p^2+2p^2 = (2p)^2$ for a string of length $4p^2+2p$ would be OK.