# Show co-prime...

• Sep 25th 2011, 06:22 AM
elim
Show co-prime...
Suppose that $a,b\in\mathbb{N}^+,\ \gcd(a,b) = 1$ and $p$ is an odd prime.

Show that $\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right) \in\{1,p\}$
• Sep 25th 2011, 07:02 AM
PaulRS
Re: Show co-prime...
Note that $a^p + b^p = (a + b) \cdot ( a^{p-1} - a^{p-2} b + a^{p-3} b^2 ... + b^{p-1})$
(because, more generally $x^n - y^n = (x-y)\cdot (x^{n-1}y^0 + x^{n-2}y^1 + ... + x^1 y^{n-2} + x^0y^{n-1})$ now substitute: $x = a, y = -b, n = p$ )

So then $\frac{a^p + b^p}{a+b} = a^{p-1} - a^{p-2} b + a^{p-3} b^2 ... + b^{p-1}$ , but now $a^{p-1} - a^{p-2} b + a^{p-3} b^2 ... + b^{p-1} \equiv a^{p-1} - a^{p-2}\cdot (-a) + ... + (-a)^{p-1} (\bmod a + b)$ since $a\equiv - b(\bmod.a+b)$

That is $\frac{a^p + b^p}{a+b} \equiv p\cdot a^{p-1} (\bmod. a + b)$ $\dots$ (Wink)
• Sep 25th 2011, 01:38 PM
elim
Re: Show co-prime...
Thanks a lot PaulRS!