prime number proof help

[shelly89]

prove that 7 is the only prime p of the form \(\displaystyle n^{3}-1\) for n in the set of natural numbers


very lost with this question,

I have factored it to

however how do i 'prove' that 7 is the only prime number that can be written in this way ?
\(\displaystyle (n^{2}+n+1)(n-1) \)

 

[Romsek]

prove that 7 is the only prime p of the form \(\displaystyle n^{3}-1\) for n in the set of natural numbers


very lost with this question,

I have factored it to

however how do i 'prove' that 7 is the only prime number that can be written in this way ?
\(\displaystyle (n^{2}+n+1)(n-1) \)

at n=2 you get n^3-1 = 7

what are the allowable factors of a prime number p?

if p = n^3 - 1, and n>2, what are the factors of p? (hint you listed at least two of them above)

Are all these factors allowable if p is to be considered prime?

 

[HallsofIvy]

prove that 7 is the only prime p of the form \(\displaystyle n^{3}-1\) for n in the set of natural numbers


very lost with this question,

I have factored it to

however how do i 'prove' that 7 is the only prime number that can be written in this way ?
\(\displaystyle (n^{2}+n+1)(n-1) \)

For what values of n is n-1= 1? Are there any values of n such that \(\displaystyle n^3+ n+ 1= 1\)?

Do you see why those questions are important?

 

[Deleted member 4993]

By similar logic,

for p = n³ + 1 → p cannot be prime when p > 2