Catalan's Identity

Author

Eric W. Weisstein
January 17, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/CatalansIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CatalansIdentity.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Identities

((((Fibonacci[n])))^2)-(Fibonacci[n+r]) (Fibonacci[n-r])==((((Fibonacci[r])))^2) (((-1))^(n-r))

v4.2

FullSimplify[Fibonacci[n]^2-Fibonacci[n+r]Fibonacci[n-r]-(-1)^(n-r)Fibonacci[r]^2,(n|r)∈Integers]

((((x^2)+(y^2)+(z^2)))^3)==(((x (-(x^2)-(y^2)+3 (z^2))))^2)+(((y (-(x^2)-(y^2)+3 (z^2))))^2)+(((z (-3 (x^2)-3 (y^2)+(z^2))))^2)

Subtract@@{(x^2+y^2+z^2)^3,(x(3z^2-x^2-y^2))^2+(y(3z^2-x^2-y^2))^2+(z(z^2-3x^2-3y^2))^2}//Expand

FullSimplify[(x(3z^2-x^2-y^2))^2+(y(3z^2-x^2-y^2))^2+(z(z^2-3x^2-3y^2))^2]