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# Prime Number Question Watch

1. This is the question and how far I've gotten;

With k being a natural number such that is a prime number (i.e 3, 7, 31) Define and list all positive integers which divide N;

So the positive integers that divide N must divide or p. Now p is prime, so the only thing that can divide it is itself and 1. The only things that can divide are multiples of two.

So divisors of are

Thus this tells me the positive integers which divide N (I'm assuming there's an infinite number of possibilities because I'm not sure if there is or isn't) are;

Now it asks prove that the sum of all its divisors (including 1 but not N) is equal to N , this bit I can't do.

It's saying

Basically simplified (since )

Now if I take a factor of out I get

Which basically means I have to show that

Sorry for the long explanation and I hope you understand. Let me know if I've gone wrong somewhere. I can't do the bolded bit, how exactly do I prove that last statement? Any help/hints appreciated.

2. It's probably (slightly) easier to try to show that the sum of all the factors, including N, sum to 2N.
3. (Original post by Hopple)

It's probably (slightly) easier to try to show that the sum of all the factors, including N, sum to 2N.
Hey thanks. Is that just a general equation I'm meant to know or something? And I'm not sure if I can do what you proposed as the question says without including N as a divisor.

I'm not exactly sure how I can use your useful piece of information. I know that

And using what you've said;

But then I just get

What have I done wrong? :|
4. Let's take a concrete example: k=3, so p = 2^k - 1 = 7. Then N = 4.7 = 28.

If you list all the factors of 28, you'll see that you've missed (almost) a whole category in your earlier analysis.
5. (Original post by Anti Elephant Mine)

Thus this tells me the positive integers which divide N (I'm assuming there's an infinite number of possibilities because I'm not sure if there is or isn't) are;

i think that you have overlooked factors made by combining the powers of 2 with the prime number p...

the bear

dfranklin beat me to it
6. (Original post by Anti Elephant Mine)
Hey thanks. Is that just a general equation I'm meant to know or something? And I'm not sure if I can do what you proposed as the question says without including N as a divisor.
It'd be handy to know, yes I'm sure you can do it that way, as long as you subtract off N at the end.

Does 2p divide N?
7. Ah my mistake, I've done it now, can't believe I didn't notice that. Thanks for the help, rep for all of you!

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Updated: December 12, 2010
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