Brocard's problem: why my approach excludes 4, 5, 7

[dimension10]

I have attatched my Paradox for the existence of 4,5 and 7 using Brocard's problem . I don't know where i have gone wrong as 4,5,7 exist, surely.

 

[atomthick]

On the second page you get $\alpha\cdot sin(\alpha\pi)\Gamma(\alpha) + 1 = x^{2}$ and that is not correct since $sin(\alpha\pi)$ are canceling each other. The correct result is
$\alpha\Gamma(\alpha) + 1 = x^{2}$

 

[dimension10]

Using Euler's reflection formula its correct.

 

[atomthick]

Yup, but you apply it twice and therefore the result should have been without the sin(α*pi)