Can you at least try to make it easy for us to read????
For how many positive integers $n\geq 2$ is $1001_n$ a prime number?
What does n/geq even mean?????
If you fix your question, I will help you out.
1001_n is a prime number in the following bases:
base 5, base 22, base 24, base 25, base 26, base 28, base 33, and base 36.
No! For example: 10015 = 1*53 + 0*52 + 0*5 + 1 = 126 which is clearly not prime! Ther same is true of the others listed.
In general we have: 1001n = n3 + 1
Now (n + 1)3 = n3 + 3n2 + 3n + 1
Rearrange this to get: n3 + 1 = (n+1)3 - 3n2 - 3n = (n+1)3 - 3n(n+1) = (n+1)( (n2 - n+1)
In other words n3 + 1 can always be expressed as the product of at least two factors, hence there are no positive integers, n, greater than or equal to 2 for which numbers of the form 1001n are prime.