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# help

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

Jan 2, 2020

### 4+0 Answers

#1
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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
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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
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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
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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