This was a surprising pair of identities I stumbled on in college.
and
So, for example, with a=10 and b=2,
I don’t remember the source of the question, but I suspect it was someone with knowledge of this relation.
I’m very curious whether this is part of a larger class of mathematical expressions involving roots that can be simplified.
Here’s a proof:
Solving for m and n:
Interestingly, this can be used in a telescoping identity when a=k, b=1:
