+0  
 
0
50
4
avatar

For how many positive integers $n\geq 2$ is $1001_n$ a prime number?

 Jan 2, 2020
 #1
avatar+1692 
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.

 Jan 2, 2020
 #2
avatar+28357 
+1

I guess this is meant to be the following:

 

\(\text{For how many positive integers }n\geq2\text{ is }1001_n\text{ a prime number?}\)
 

I presume the subscript n indicates a number base. 

Alan  Jan 3, 2020
 #3
avatar
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.

 Jan 3, 2020
 #4
avatar+28357 
+1

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.

Alan  Jan 4, 2020

20 Online Users

avatar