Number of Integer Solutions

The equation $a^b = 1$ has solutions in three cases: Case 1: $b = 0$ and $a \neq 0$. $x^2 - 3x - 10 = 0 \implies (x - 5)(x + 2) = 0 \implies x = 5$ or $x = -2$. For $x = 5$: $a = 25 - 10 = 15 \neq 0$. For $x = -2$: $a = 4 - 10 = -6 \neq 0$. Solutions: $x = 5, -2$.

Case 2: $a = 1$. $x^2 - 10 = 1 \implies x^2 = 11$, no integer solution.

Case 3: $a = -1$ and $b$ is an even integer. $x^2 - 10 = -1 \implies x^2 = 9 \implies x = 3$ or $x = -3$. For $x = 3$: $b = 3^2 - 3(3) - 10 = -10$ (even). For $x = -3$: $b = (-3)^2 - 3(-3) - 10 = 9 + 9 - 10 = 8$ (even). Solutions: $x = 3, -3$.

Total integer solutions = $\{5, -2, 3, -3\}$, which gives 4 solutions.