If $a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q$ .prove that $ a=b=c$ or $p=q=r$?

Question

Let $$a,b,c,p,q,r$$ be positive integers such that :

$$a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q$$

How do I prove :$ a=b=c$ or $p=q=r$ ?

Thank you for any kind of help.

Answer 1

The case when all of $a,b,c$ are not distinct is straightforward, as well as the cases when not all of $p,q,r$ are different.

Assume $a>b>c$ and assume $p> q,r$.

Case $1$ is when $p>q> r$. In this case clearly $a^p+b^q+c^r$ is strictly larger than any possible combination.

Case $2$ is when $p>r>q$

In this case clearly $a^q+b^r+c^p$ is smaller than any other combination.