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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?
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?