Question

Show that the the only prime of form n^3-1 is 7.

Answer 1

n^(3-1) or n^(3)-1?

Answer 2

\[n^3-1\]

Answer 3

because it factors

Answer 4

n^3-1=
n^3-1^3=
(n-1)(n^2-n+1)

We know:
If n is an integer, n-1 is an integer and n^2-n+1 is also an integer.
Thus,
(n-1)(n^2-n+1)=ab for integers a and b
if ab is prime, either a or b must be 1 (or else, we would have more than two factors).
Thus, either:
n-1=1
OR
n^2-n+1=1
If the first is true:
n=2
If the second is true,
n=0 and n=1
So, we test n=0, n=1, and n=2 in n^3-1 to see which one(s) are prime:
n=0, n^3-1=0-1=-1 not prime
n=1, n^3-1=1-1=0 not prime
n=2, n^3-1=8-1=7 prime

Thus, 7 is the only prime in the form of n^3-1.