I How do you Express A and B in q and r?

[heff001]

A - B - q^n + r^n = 0, (A, B) co prime, A odd, B even, n odd, gcd (A / q, q) = 1, gcd (B / r, r) = 1

Problem
A - B - q^n + r^n = 0,
(A, B) co prime,
A odd,
B even,
n odd,
gcd (A / q, q) = 1,
gcd (B / r, r) = 1.

 

[Office_Shredder]

I'm not sure about a total set of solutions, but $n=1$, $A=q$ and $B=r$ is one solution.

It's not explicitly stated, but since $A$is divisible by $q$ and $B$ is divisible by $r$, $q$ and $r$ are coprime as well.

 

[heff001]

You did great !