Are the only solutions to $a^n + b^n =(a+b \forall n\geq 1$, 0, 1?

Question

Suppose $a^n + b^n = a+b$ for all $n \in \mathbb{N}$.

Is it true that the only solutions are:

$$a=1, b=0$$ $$a=0, b=1$$ $$a=b=0$$

?

If so, how do you prove it?

Answer 1

Let $A:=\{(1,0),(0,1),(0,0),(1,1)\}$

1. It is clear that if $(a,b) \in A$, then $a^n+b^n=a+b$.

2. If $(a,b)=(1,-1)$ or $=(-1,1)$, then $a^2+b^2=2 \ne 0=a+b$.

3. Let $a^n+b^n=a+b$ for all $n$. If $|a|>1$ or $|b|>1,$ then the sequence $(a^n+b^n)$ is unbounded, a cotradiction , since this sequence is constant.

Hence, by 1., 2. and 3. , we have only to investigate the case $|a| < 1$ and $|b| < 1$.

From $a^n+b^n=a+b$ for all $n$ we get with $ n \to \infty$ that $ a=-b$.

Hence $0=a+b=a^2+a^2=2a^2$, therefore $a=b=0$.

Conclusion: $a^n+b^n=a+b \iff (a,b) \in A$.