I Wonder How Many Solutions Exist

Both terms on the left-hand side are positive. We must have $139y^8 \le 1553$ , so $y^8 \le \frac{1553}{139} = 11.17\ldots$ , so the only possible values of $y^8$ for integer $y$ are $0$ or $1$ .

On substituting in, neither of these give an integer value for $x$ , so there are no solutions.

the number of solutions should be a multiple of $4$ . How about that? :D

Since $x^4 \equiv 0,1 \pmod{5}$ we deduce that $x^4 + 139y^8 \equiv 1,0,-1 \pmod{5}$ . Since $1553 \equiv 3 \pmod{5}$ there can be no solutions to the equation.

$a^{16} \equiv 1 \pmod{17} \implies x^{4} \in \left\{-4, -1, 0, 1, 4 \right\}, y^{8} \in \left\{-1, 0, 1 \right\} \text{ when looking mod 17} \\ \text{ Looking mod 17 gives } x^{4} + 3y^{8} \equiv 6 \pmod{17} \text{ and } x^{4} + 3y^{8} \in \left\{-7, -4, -3, -2, -1, 0, 1, 2, 3, 4, 7 \right\} \\ 6\notin \left\{-7, -4, -3, -2, -1, 0, 1, 2, 3, 4, 7 \right\} \text{ so there are no integer solutions. }$