The only prime of the form n^3-1 is 7?

In summary, the only prime of the form n^3-1 is 7. This is proven by writing n^3-1 as (n-1)(n^2+n+1) and noting that for any natural number n, n^2+n+1 is always greater than 1. Therefore, the only possible value for n-1 is 1, which results in n=2 and p=7.
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Math100
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Homework Statement
Prove the assertion below:
The only prime of the form n^3-1 is 7.
[Hint: Write n^3-1 as (n-1)(n^2+n+1).]
Relevant Equations
None.
Proof: Suppose p is a prime such that p=n^3-1.
Then we have p=n^3-1=(n-1)(n^2+n+1).
Note that prime number is a number that has only two factors,
1 and the number itself.
Since n^2+n+1>1 for ##\forall n\in\mathbb{N}##,
it follows that n-1=1, and so n=1+1=2.
Thus n=2, and so p=n^3-1=2^3-1=8-1=7.
Therefore, the only prime of the form n^3-1 is 7.

Above is my proof for this assertion. Can anyone please verify/review it and see if it's correct?
 
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Math100 said:
Homework Statement:: Prove the assertion below:
The only prime of the form n^3-1 is 7.
[Hint: Write n^3-1 as (n-1)(n^2+n+1).]
Relevant Equations:: None.

Above is my proof for this assertion. Can anyone please verify/review it and see if it's correct?
It is correct, and your wording is improving!
 
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  • #3
fresh_42 said:
It is correct, and your wording is improving!
Thank you!
 

FAQ: The only prime of the form n^3-1 is 7?

1. What is the significance of the statement "The only prime of the form n^3-1 is 7?"

The statement refers to a mathematical theorem known as the "Mersenne prime theorem," which states that the only prime number of the form n^3-1 is 7. This theorem has important implications in number theory and has been studied by mathematicians for centuries.

2. How was this theorem proven?

The proof of this theorem involves a combination of algebraic manipulations and number theory concepts. The original proof was given by mathematician Leonhard Euler in the 18th century, but there have been numerous variations and extensions of the proof since then.

3. Are there any exceptions to this theorem?

No, there are no known exceptions to this theorem. However, it is important to note that this theorem only applies to prime numbers of the form n^3-1, and there are many other prime numbers that do not fall under this category.

4. What are some real-world applications of this theorem?

One possible application of this theorem is in cryptography, where prime numbers are used to create secure codes and passwords. Additionally, this theorem has implications in the study of perfect numbers and Mersenne primes, which have practical applications in computer science and engineering.

5. Can this theorem be extended to other forms, such as n^3+1?

No, this theorem only applies to the specific form n^3-1. However, there are other theorems and patterns that have been discovered for different forms of prime numbers, and further research is being conducted in this area.

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