: Mathématiques: Candido's identity

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Candido's identity

Candido's identity, named after the Italian mathematician Giacomo Candido, is an identity for real numbers. It states that for two arbitrary real numbers ${\\displaystyle x}$ and ${\\displaystyle y}$ the following equality holds:
${\\displaystyle \\left[x^{2}+y^{2}+(x+y)^{2}\ ight]^{2}=2[x^{4}+y^{4}+(x+y)^{4}]}$
The identity however is not restricted to real numbers but holds in every commutative ring. Candido originally devised the identity to prove the following identity for Fibonacci numbers:
${\\displaystyle (f_{n}^{2}+f_{n+1}^{2}+f_{n+2}^{2})^{2}=2(f_{n}^{4}+f_{n+1}^{4}+f_{n+2}^{4})}$

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