$t^2=at+b$ where $t$ is real and $a,b$ are +ve integers. Show that $t^3$ is not equal to $8t+5$.

Question

$t^2=at+b$ where $t$ is real and $a,b$ are +ve integers. Show that $t^3$ is not equal to $8t+5$.

After solving for $t$ in terms of $a, b$ could not advance any further.

Answer 1

$$t^3 = t(t^2) = t(at+b) = at^2 + bt = a(at+b) + bt = (a^2 +b)t + ab.$$ Now all that remains is to show that the system $$a^2 + b = 8, \\ ab = 5,$$ has no positive integer solutions. Clearly, $ab = 5$ implies $\{a, b\} = \{1, 5\}$ in some order. So you have only two cases to check.