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 [itex]\alpha\cdot sin(\alpha\pi)\Gamma(\alpha) + 1 = x^{2}[/itex] and that is not correct since [itex]sin(\alpha\pi)[/itex] are canceling eachother. The correct result is
[itex]\alpha\Gamma(\alpha) + 1 = x^{2}[/itex]

*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)

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atomthick	On the second page you get [itex]\alpha\cdot sin(\alpha\pi)\Gamma(\alpha) + 1 = x^{2}[/itex] and that is not correct since [itex]sin(\alpha\pi)[/itex] are canceling eachother. The correct result is [itex]\alpha\Gamma(\alpha) + 1 = x^{2}[/itex]

atomthick	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)