Vorobiev's theorem states that if 
, then 
, where 
is a Fibonacci number
and 
means 
divides 
. The theorem was discovered by Vorobiev in 1942, but not published
until 1967. It was used by Y. Matiyasevich in his negative solution to the Hilbert's
tenth problem.
Note that the converse does not hold. For example, 
, but 
. The plot above shows values of 
for which 
and 
(black) and for which 
but 
(red).
