#1**0 **

Guest-

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.

tommarvoloriddle Jan 2, 2020

#3**0 **

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.

Guest Jan 3, 2020

#4**+1 **

No! For example: 1001_{5} = 1*5^{3} + 0*5^{2} + 0*5 + 1 = 126 which is clearly not prime! Ther same is true of the others listed.

In general we have: 1001_{n} = n^{3} + 1

Now (n + 1)^{3} = n^{3} + 3n^{2} + 3n + 1

Rearrange this to get: n^{3} + 1 = (n+1)^{3} - 3n^{2} - 3n = (n+1)^{3} - 3n(n+1) = (n+1)( (n^{2} - n+1)

In other words n^{3} + 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 1001_{n} are prime.

Alan
Jan 4, 2020