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

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In summary, Brocard's problem is a mathematical paradox that presents a situation where there are an infinite number of possibilities, but only a finite number of solutions. This is related to the existence of 4, 5, and 7 in the problem, as these numbers have unique properties that make them the only values for n that result in a solution. This paradox challenges our understanding of mathematics and highlights the limitations of our current knowledge and techniques. It is also similar to other famous paradoxes in mathematics, such as the Collatz conjecture and Goldbach's conjecture.
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.

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• the nonexistence of {4, 5, 7}.pdf
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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 each other. The correct result is
$\alpha\Gamma(\alpha) + 1 = x^{2}$

Using Euler's reflection formula its correct.

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

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.

## 1. How does Brocard's problem relate to the existence of 4, 5 and 7 in a paradox?

Brocard's problem is a mathematical problem that involves finding integer solutions to the equation n! + 1 = m^2, where n and m are integers. This problem is related to the existence of 4, 5, and 7 in a paradox because it has been shown that the only solutions to this equation are when n equals 4, 5, or 7. This creates a paradox because there are an infinite number of possible values for n, but only three of them result in a solution.

## 2. Why is Brocard's problem considered a paradox?

Brocard's problem is considered a paradox because it presents a situation where there are an infinite number of possibilities, but only a finite number of solutions. This goes against our intuition and understanding of mathematics, where we expect there to be a solution for every possible input. It also highlights the limitations of our current mathematical knowledge and techniques.

## 3. How is the existence of 4, 5, and 7 in Brocard's problem explained?

The existence of 4, 5, and 7 in Brocard's problem can be explained by the unique properties of these numbers. For example, 4 is the only number that is both a square number and a factorial number (4! = 24). Similarly, 5 is the only number that is both a square number and a prime number. These special properties make them the only values for n that result in a solution to the equation n! + 1 = m^2.

## 4. What implications does Brocard's problem have for our understanding of mathematics?

Brocard's problem challenges our understanding of mathematics and highlights the limitations of our current knowledge and techniques. It shows that there are still unsolved problems and paradoxes in mathematics that require further investigation and exploration. It also raises questions about the relationship between numbers and their properties, and how we define and understand them.

## 5. How does Brocard's problem relate to other paradoxes in mathematics?

Brocard's problem is just one example of a paradox in mathematics where there are an infinite number of possibilities, but only a finite number of solutions. Other famous examples include the Collatz conjecture and the Goldbach's conjecture. These paradoxes challenge our understanding of mathematics and highlight the need for further research and exploration in the field.

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