If $(a,b)=1$ and $n$ is a prime, prove that $(a^n+b^n)/(a+b)$ and $(a+b)$ have no common factor, unless $a+b$ is a multiple of $n$

Question

I have no hints to get started. This is a really tough problem involving prime powers and it's the first time I have seen such question.

I need some good hints to get started. Thanks!

Please do not use congruence, limits or derivatives because they are out of scope of the chapter which has this question.

Answer 1

Assume $a+b$ is not a multiple of $n$.

For $n=2$, a common prime factor of $a+b$ and $\frac{a^2+b^2}{a+b}=a+b-\frac{2ab}{a+b}$ must be $2(=n)$, which is excluded, or must divide one of $a,b$ and hence both. Thus we may assume that $n$ is odd.

Let $q$ be a prime with $q^k\| a+b$, $k\ge 1$. So $q\nmid n$ and we have $b=-a+rq^k$ with $q\nmid r$. We need to show that $q^{k+1}\nmid a^n+b^n$: $$a^n+b^n=a^n+(-a)^n+n(-a)^{n-1}rq^k+{n\choose 2}(-a)^{n-2}r^2q^{2k}+\ldots\equiv na^{n-1}rq^k\pmod{q^{k+1}}$$ So we want to show that $q\nmid na^{n-1}r$. We already know $q\nmid r$ and $q\nmid n$. Remains the possibility that $q\mid a$, but then also $q\mid (a+b)-a=b$, contradicting $\gcd(a,b)=1$. Hence indees $q\nmid na^{n-1}r$, $q^{k+1}\nmid a^n+b^n$ and ultimately $q\nmid \frac{a^n+b^n}{a+b}$.

Answer 2

well , the problem may seem confusing but it can be easily approached using fermat theorem as, a^n==a mod n .....(1) and b^n== b mod n ....(2) now, adding both a^n+b^n==a+b mod n now if (a+b) is a multiple of n, a^n+b^n==0 mod n, dividing by (a+b) on both the sides we can imply that , a^n+b^n/a+b and a+b have no common factors unless a+b is a multiple of n thanks...waiting for response!