How to prove that there is a function index t so that t³ + t + 1 is an index of the same function? I would like to prove $\exists t\phi_t = \phi_{t^3+t+1}$ where $\phi_0,\phi_1,\phi_2,...$ are sequence of all of the partially computable function.

$\phi_t = \phi_{t^3+t+1}$ only if $t = t^3+t+1$ and it means $t=-1$! and there is no program with number $-1$!
This is just a special case of Rogers' fixed-point theorem which says that, for any total computable function&nbsp;$f$, there exists&nbsp;$t$ such that $\phi_t = \phi_{f(t)}$.