5. Find all non-negative integer solutions $(x, y)$ to the equation $2^{x}=5^{y}+3$.

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5. The non-negative integer solutions satisfying $x \leqslant 7$ are
$$
(x, y)=(2,0),(3,1),(7,3) \text {. }
$$

Assume $x \geqslant 8$.
Taking modulo $2^{8}$ on both sides of the original equation, we get
$$
5^{y} \equiv-3\left(\bmod 2^{8}\right) \text {. }
$$

Notice that, $5^{3}=2^{7}-3$, and the order of 5 modulo $2^{7}$ and modulo $2^{8}$ are 32 and 64, respectively.
Thus, $y \equiv 3(\bmod 32), y \not \equiv 3(\bmod 64)$.
Therefore, $y \equiv 35(\bmod 64)$.
Taking modulo 257 on both sides of the original equation, we get
$$
\begin{array}{l}
2^{x} \equiv 5^{y}+3(\bmod 257) . \\
\text { By } 5^{35} \equiv 14(\bmod 257), \\
5^{64 k} \equiv \pm 1, \pm 16(\bmod 257),
\end{array}
$$

we have $5^{y}+3 \equiv-30,-11,17,36(\bmod 257)$.
However, $2^{x}$ can only be congruent to $\pm 1, \pm 2$, $\pm 4, \cdots, \pm 128$ modulo 257, so there are no solutions for $x \geqslant 8$.

Therefore, the non-negative integer solutions to the original equation are only $(2,0)$, $(3,1)$, and $(7,3)$.