Simplifying non-trivial root expressions

Jun 9, 08:01 AM

This was a surprising pair of identities I stumbled on in college.

167be10ac38834e0d3493cce6841d7646b9dfccd

and

c3baaa8b26e39aaf81dcc9ee0492cfd05857ce06

So, for example, with a=10 and b=2, 8cf03a9fc049419d85c28ec95b88054299ae0cc1

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:
f71b19cdb2dc1af8331e0cdd32dd7c4c80c31a5e
 
6ba189a5f9bc88eb54175d2c6eba285a9cec4e32
 
8956774fb7cebd1affe297b1e72b49802c50de97
 
cff6bd32c9a48d7826683e2d4753cfd92a71ba66
 
1021c6b5adb867a12775ed18b10c010b9408f1d5
 
80fce0f5fa5ebdd3857726668eb3441d6f69f815

Solving for m and n:

a77a14002b55328d044275c83d7b764b70eebdbb   e3d25a3f776fce9a4d1b73a79ee60b686d5450c9

Interestingly, this can be used in a telescoping identity when a=k, b=1:

6ae782b9bb859f9597c83626a69f1e5b9e8702e0
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