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

Question

I assumed the GCD of both numbers is $d$ and it's evident that $d|a+b$ and $d|a^n+b^n$

But now I don't have any idea how to move further.Any suggestions?? Note : this question is asked from a elementary guide to number theory.so I highly recommend to use only elementary properties of GCD to prove

For odd $n,$ $$b^n=-(-b)^n$$ https://math.stackexchange.com/questions/247146/show-that-gcd-left-fracan-bna-b-a-b-right-gcdn-dn-1-a-b — lab bhattacharjee, Nov 30 '19 at 14:36

Angina Seng · Answer 1 · 2019-11-30T15:24:24.233

1

Let $p$ be an odd prime. Let $c=a+b$ and $$d=\frac{a^p+b^p}{a+b}=a^{p-1}-a^{p-2}b+a^{p-3}b^2-\cdots+b^{p-1}.$$ If $q\ne p$ is a prime factor of both $c$ and $d$, then $b\equiv -a\pmod q$ and so $$0\equiv d\equiv a^{p-1}+a^{-p-1}+\cdots+a^{p-1}=pa^{p-1}\pmod q.$$ Then $a\equiv0\pmod q$, and $b\equiv-a\equiv0\pmod q$ contradicting $\gcd(a,b)=1$.

edited Nov 30 '19 at 15:24

answered Nov 30 '19 at 14:39

Angina Seng

158,341

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

1 Answers1