Vorobiev's Theorem

Author

Eric W. Weisstein
December 30, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/VorobievsTheorem.nb.

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

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

Check

pairs=Flatten[Outer[List,Range[15],Range[15]],1];

mod1=Select[pairs,Mod[Fibonacci[#[[1]]],Fibonacci[#[[2]]]^2]==0&]

mod2=Select[pairs,Mod[#[[1]],Fibonacci[#[[2]]]]==0&]

Complement[mod1,mod2]

Complement[mod2,mod1]

ArrayPlot[Table[If[Mod[Fibonacci[k],Fibonacci[l]^2]==0,1,0]+If[Mod[k,Fibonacci[l]]==0,2,0],{l,15,1,-1},{k,120}],
ColorRules->{3->Black,2->Red,1->Blue,0->White},
ImageSize->500,FrameLabel->TraditionalForm/@{k,l}]

Counterexample to Converse

Fibonacci[2]/Fibonacci[3]^2

2/Fibonacci[3]