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

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.
Attached Files
 the nonexistence of {4, 5, 7}.pdf (176.5 KB, 64 views)
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 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 eachother. The correct result is $\alpha\Gamma(\alpha) + 1 = x^{2}$
 Using Euler's reflection formula its correct.

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

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