Algebraic Identity

Author

Eric W. Weisstein
April 13, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/AlgebraicIdentity.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Ferrari's Identity

Expand[Subtract@@{(a^2+2a c-2b c-b^2)^4+(b^2-2a b-2a c-c^2)^4+(c^2+2a b+2b c-a^2)^4,2(a^2+b^2+c^2-a b+a c+b c)^4}]

Fibonacci Identity

Simplify[Subtract@@{(a^2+b^2)(c^2+d^2),(a c-b d)^2+(b c+a d)^2}]

Kohmoto Identity

poly=(1679616a^16-66096a^10b^6+153a^4b^12)^3+(-1679616a^16-559872a^13b^3-27216a^10b^6+3888a^7b^9+63a^4b^12-3a b^15)^3+(1679616a^15b+279936a^12b^4-11664a^9b^7-648a^6b^10+9a^3b^13+b^16)^3

Expand[poly]

Lebesgue Identity

((((a^2)+(b^2)+(c^2)+(d^2)))^2)=(((2 a d-2 b c))^2)+(((2 a c+2 b d))^2)+((((a^2)+(b^2)-(c^2)-(d^2)))^2)

Subtract@@{(a^2+b^2+c^2+d^2)^2,(a^2+b^2-c^2-d^2)^2+(2a c+2b d)^2+(2a d-2b c)^2}//Simplify

Liouville Polynomial Identity

<<DiscreteMath`Combinatorica`

t=KSubsets[Array[x,4],2]

Expand[Subtract@@{Plus@@(Plus@@@t^4)+Plus@@(Subtract@@@t^4),6Plus@@(Array[x,4]^2)^2}]

Ramanujan 6-10-8 Identity

Expand[Subtract@@{
64((a+b+c)^6+(b+c+d)^6-(c+d+a)^6-(d+a+b)^6+(a-d)^6-(b-c)^6)((a+b+c)^10+(b+c+d)^10-(c+d+a)^10-(d+a+b)^10+(a-d)^10-(b-c)^10),45((a+b+c)^8+(b+c+d)^8-(c+d+a)^8-(d+a+b)^8+(a-d)^8-(b-c)^8)^2
}/.d->b c/a]