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)

 

[blue_raver22]

I can understand the confusion and apparent paradox in the existence of 4, 5, and 7 in relation to Brocard's problem. However, it is important to note that Brocard's problem is a mathematical problem and not a statement about the existence of numbers. In mathematics, we often use hypothetical or imaginary scenarios to explore and understand concepts and principles.

In the case of Brocard's problem, we are given a hypothetical scenario where two prime numbers, P and P+2, can be expressed as the factorial of two other numbers, n! and (n+1)!. This scenario is not meant to be a statement of fact, but rather a thought experiment to explore the concept of factorials and prime numbers.

Furthermore, the numbers 4, 5, and 7 do exist in mathematics and in the real world. They are well-defined and have been studied extensively in various fields, including number theory and algebra. Therefore, it is incorrect to say that these numbers do not exist.

In conclusion, while Brocard's problem may seem paradoxical in relation to the existence of 4, 5, and 7, it is important to understand that it is a mathematical problem and not a statement about the existence of numbers. These numbers do exist and have been studied extensively in mathematics.