Context: Brocard's problem is a problem in mathematics that asks to find integer values of $n$ and $m$ for which$$ n!+1 = m^2 \tag{1}$$ Let's define, $$T=\left(\left\lfloor \frac{ (\lfloor\log(n) \rfloor -1)-1}{2}\right\rfloor +(n-2^{ \lfloor\log(n) \rfloor}+1) \times \lfloor\log(n) \rfloor \right)$$
Using elementary argument, it can be shown that,
Brocard's problem has no solution when $T$ is odd.
Problem: I would like to prove it using induction method.
Question: How should I approach the problem?
It would be nice if someone post the proof. If that is not possible, leave a direction, please.