This is a most surprising and amazing identity from the 1965 Polish Mathematical Olympiads.

“31. Prove that if *n* is a natural number, then we have

(√2 – 1)* ^{n}* = √

*m*– √(

*m*– 1),

where *m* is a natural number.”

Here, natural numbers are 1, 2, 3, …

I found it to be quite challenging, as all the Polish Math Olympiad problems seem to be.

See the Amazing Identity