Candido's Identity

Giacomo Candido (1871-1941) has invented a formula that seems to have been published posthumously ten years after his death:

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

The formula can be easily verified by carrying through the operations involved. There is also a proof without words by Alsina and Nelsen:

Candido's formula - a proof without words

Candido applied the formula to establishing a property of the Fibonacci numbers:

\((F_{n-1}^{2}+F_{n}^{2}+F_{n+1}^{2})^{2}=2(F_{n-1}^{4}+F_{n}^{4}+F_{n+1}^{4})\).

The formula is little known - I found it mentioned only in a recent book by R. Grimaldi.

Alsina and Nelsen generalized the question by asking for solutions \(f\) of the functional equation

\(f(f(x) + f(y) + f(x + y)) = 2 [f(f(x)) + f(f(y)) + f(f(x + y))].\)

They proved that, assuming \(f\) is continuous function from \([0,\infty)\) onto \([0,\infty)\) such that \(f(0) = 0\), the solution is unique (up to a constant factor), \(f(x)=x^2\). There are really great many discontinuous solutions. Any \(f\) with the property that \(f(x) =0\) for rational \(x\) and \(f(x)\) rational (but arbitrary!), otherwise, satisfies the generalized equation!

R. S. Melham extended the property of the Fibonacci numbers by showing (among others) that

\(2(F_{n}^{2}+F_{n+1}^{2}+F_{n+2}^{2}+F_{n+3}^{2})^{2}=3(F_{n}^{4}+F_{n+1}^{4}+F_{n+2}^{4}+F_{n+3}^{4})^{4}\).

References

  1. C. Alsina, R. B. Nelsen, Candido's Identity, Mathematics Magazine, Vol. 78, No. 2 (Apr., 2005), p. 131
  2. C. Alsina, R. B. Nelsen, On Candido's Identity, Mathematics Magazine, Vol. 80, No. 3 (Jun., 2007), pp. 226-228
  3. G. Candido, A Relationship Between the Fourth Powers of the Terms of the Fibonacci Series, Scripta Mathematica, 17:3-4 (1951) 230
  4. R. Grimaldi, Fibonacci and Catalan Numbers: an Introduction, Wiley, 2012
  5. R. S. Melham, YE OLDE FIBONACCI CURIOSITY SHOPPE REVISITED, Fibonacci Quarterly, 2004, 2, 155-160

Fibonacci Numbers

  1. Ceva's Theorem: A Matter of Appreciation
  2. When the Counting Gets Tough, the Tough Count on Mathematics
  3. I. Sharygin's Problem of Criminal Ministers
  4. Single Pile Games
  5. Take-Away Games
  6. Number 8 Is Interesting
  7. Curry's Paradox
  8. A Problem in Checker-Jumping
  9. Fibonacci's Quickies
  10. Fibonacci Numbers in Equilateral Triangle
  11. Binet's Formula by Inducion
  12. Binet's Formula via Generating Functions
  13. Generating Functions from Recurrences
  14. Cassini's Identity
  15. Fibonacci Idendtities with Matrices
  16. GCD of Fibonacci Numbers
  17. Binet's Formula with Cosines
  18. Lame's Theorem - First Application of Fibonacci Numbers

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