Does there exist $a,b,c\in \mathbb{R}$ such that for all $n$, $a^{2n+1}+b^{2n+1}=c^{2n+1}$?

Question

Let $a,b,c \in \mathbb{R}$, $n \in \mathbb{N}$

Is it possible that there are a,b,c that fulfill the following equation for every $(2n+1)$?

$a^{2n+1}+b^{2n+1}=c^{2n+1}$

$(i.e \quad a^3+b^3=c^3, a^5+b^5=c^5......)$

Answer 1

Let $a,b,c$ satisfy the above equation for all $n$. We have $$ c =(a^{2n+1} + b^{2n+1})^{\frac{1}{2n+1}}$$ If this holds for every $n$, it holds in limit, and so by a fact about $L^p$-norms $c = \sup \{a,b\}$. Then, say $c=a$ WLOG. Then, for the property to hold for $n=1$ we need $b = 0$.