For a natural number $$n$$, let $$\alpha_n = 19^n - 12^n$$. Then, the value of $$\frac{31\alpha_9 - \alpha_{10}}{57\alpha_8}$$ is ______.

We have $$\alpha_n = 19^n - 12^n$$.

By expanding the numerator we get $$ 31\alpha_9 - \alpha_{10} = 31(19^9 - 12^9) - (19^{10} - 12^{10}) $$
$$ = 31 \cdot 19^9 - 31 \cdot 12^9 - 19^{10} + 12^{10} $$
$$ = 19^9(31 - 19) - 12^9(31 - 12) $$
$$ = 12 \cdot 19^9 - 19 \cdot 12^9 $$
$$ = 12 \cdot 19(19^8 - 12^8) = 12 \cdot 19 \cdot \alpha_8 $$.

Meanwhile, the denominator becomes $$57\alpha_8 = 3 \cdot 19 \cdot \alpha_8$$.

Hence, the fraction simplifies as $$ \frac{31\alpha_9 - \alpha_{10}}{57\alpha_8} = \frac{12 \cdot 19 \cdot \alpha_8}{3 \cdot 19 \cdot \alpha_8} = \frac{12}{3} = 4 $$.

Therefore, the answer is $$4$$.