Please help me solve this:

Show that if gcd(a,b)=1, and p is an odd prime, then

gcd( a+b , {(a^p)+(b^p)}/(a+b) ) = 1 or p

I've tried obvious ways to show it, and even expanded by binomial theorem, but of no use.

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- Oct 19th 2011, 01:25 AM #1

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- Oct 21st 2011, 05:12 AM #2
## Re: Help with g.c.d.

First to type your question in LaTeX:

Show that if , and is an odd prime, then or .

I think that one route to try comes from the observation that the GCD of and divides any "linear combination" of them; that is, it will divide any number of the form for any integers . So, in particular, must divide

The nice part is that this shaves off and from the numerator and leaves you with terms that look like where ranges between and . And in particular, you may know that this kind of term is divisible by , which seems helpful for your statement.

Maybe you can play around with this for a while and see where it goes.

- Oct 21st 2011, 06:22 AM #3

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- Oct 22nd 2011, 01:50 PM #4

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## Re: Help with g.c.d.

Simple notation step helps. Write , then the Binomial Theorem gives (note that the last term of is ).

Divide both sides by ; hence .

Then factor out , by doing so we arrive at the following form: for some integer .

It's important to see that (explain this) and so .

Let . Because and it's true that . By definition, must divide . But from and we conclude that .

Since is prime or .

- Oct 24th 2011, 04:23 AM #5

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- Oct 24th 2011, 04:24 AM #6

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