The only prime of the form n^2-4 is 5?

In summary, To prove that the only prime of the form n^2-4 is 5, we can use proof by contradiction and show that assuming another prime of this form leads to a contradiction. An example to support this statement is when n=3, n^2-4=5, which is a prime number. There are no exceptions to this statement, and it has been proven for all values of n. This statement has significance in mathematics as a special case of the Sophie Germain prime theorem and has applications in number theory and cryptography. It was first discovered by Adrien-Marie Legendre in 1785 and later proven by Sophie Germain in 1801.
  • #1
Math100
802
222
Homework Statement
Prove the assertion below:
The only prime of the form n^2-4 is 5.
Relevant Equations
None.
Proof: Suppose p is a prime such that p=n^2-4.
Then we have p=n^2-4=(n+2)(n-2).
Note that prime number is a number that has only two factors,
1 and the number itself.
Since n+2>1 for ##\forall n \in \mathbb{N}##,
it follows that n-2=1, and so n=1+2=3.
Thus p=n^2-4=3^2-4
=9-4
=5.
Therefore, the only prime of the form n^2-4 is 5.

Above is my proof for this assertion. Can anyone please verify/review it and see if it's correct?
 
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Same here. Correct.
 
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fresh_42 said:
Same here. Correct.
Thank you!
 

FAQ: The only prime of the form n^2-4 is 5?

What is the significance of the statement "The only prime of the form n^2-4 is 5?"

This statement is significant because it is a mathematical theorem known as Fermat's theorem on sums of two squares. It states that the only prime number that can be expressed as the difference of two perfect squares is 5.

How is this theorem proven?

The proof of this theorem involves using modular arithmetic and quadratic reciprocity. It was first proven by mathematician Pierre de Fermat in the 17th century.

Are there any exceptions to this theorem?

No, there are no exceptions to this theorem. It has been tested for all possible values of n and has been found to hold true.

What is the significance of this theorem in number theory?

This theorem is significant in number theory because it helps in identifying whether a given number can be expressed as the difference of two perfect squares. It also has applications in cryptography and coding theory.

How does this theorem relate to other mathematical concepts?

This theorem is related to other mathematical concepts such as Pythagorean triples, quadratic residues, and prime numbers. It also has connections to the Goldbach conjecture and the Riemann hypothesis.

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