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Giacomo Candido devised his eponymous identity to prove that
\( {\displaystyle (F_{n}^{2}+F_{n+1}^{2}+F_{n+2}^{2})^{2} =2(F_{n}^{4}+F_{n+1}^{4}+F_{n+2}^{4})} \)

where \( F_{n} \)is the nth Fibonacci number.
The identity of Candido is that, for all real numbers x and y,[


\( {\displaystyle (x^{2}+y^{2}+(x+y)^{2})^{2} =2(x^{4}+y^{4}+(x+y)^{4})} \)

It is easy to prove that the identity holds in any commutative ring.

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