| Abstract:
The Cassini identity \( F(n-1)F(n+1) - F(n)^2 = (-1)^n \)
for the Fibonacci
sequence of numbers is well-known: in the popular math culture, it is very
often used to "prove" that, say \( 441 = 442\) or \( 64 = 65\) etc.
I have made
numerous "Math Open House" display posters and wooden cutouts based upon
this single theme. What is not that well-known is the fact that this very
same identity played a crucial role in proving the unsolvability of the
Hilbert's Tenth Problem. During the late 1960's Julia Robinson proved that
the existence of a particular Diophantine sequence of numbers would prove
the unsolvability of the Hilbert's tenth problem. In the early 1970's,
Yuri Matiyasevich (at the age of 22) used the Cassini Identity to give a
Diophantine representation of the Fibonacci function
\( n = F(m) \), that is, a
polynomial whose integer solutions are the ordered pairs
\( (m, n) \) for which \( n= F(m) \).
This, coupled with Julia Robinson's previous observation, settled
Hilbert's Tenth Problem once and for all.
-
Y. V. Matiyasevich, Hilbert's Tenth Problem, MIT Press, Cambridge,
Mass., U.S.A., 1993.
|
