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Morgado Identity

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There are several results known as the Morgado identity. The first is

F_nF_(n+1)F_(n+2)F_(n+4)F_(n+5)F_(n+6)+L_(n+3)^2=[F_(n+3)(2F_(n+2)F_(n+4)-F_(n+3)^2)]^2,
(1)

where F_n is a Fibonacci number and L_n is a Lucas number (Morgado 1987, Dujella 1995).

A second Morgado identity is satisfied by generalized Fibonacci numbers w_n,

4w_nw_(n+1)w_(n+2)w_(n+4)w_(n+5)w_(n+6)+e^2q^(2n)(w_nU_4U_5-w_(n+1)U_2U_6-w_nU_1U_8)^2 =(w_(n+1)w_(n+2)w_(n+6)+w_nw_(n+4)w_(n+5))^2,
(2)

where

e=pab-qa^2-b^2
(3)
U_n=w_n(0,1;p,q)
(4)

(Morgado 1987, Dujella 1996).

See also

Fibonacci Number, Generalized Fibonacci Number

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References

Dujella, A. "Diophantine Quadruples for Squares of Fibonacci and Lucas Numbers." Portugaliae Math. 52, 305-318, 1995.Dujella, A. "Generalized Fibonacci Numbers and the Problem of Diophantus." Fib. Quart. 34, 164-175, 1996.Morgado, J. "Note on Some Results of A. F. Horadam and A. G. Shannon Concerning a Catalan's Identity on Fibonacci Numbers." Portugaliae Math. 44, 243-252, 1987.

Referenced on Wolfram|Alpha

Morgado Identity

Cite this as:

Weisstein, Eric W. "Morgado Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MorgadoIdentity.html

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