Paradox for the existence of 4,5 and 7 using Brocard's problem

[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)