Relationship between a Fibonacci number and that closest prime number.

1 Prediction

For any given Fibonacci number, the difference between the Fibonacci number and the nearest prime number is expressed in Zeckendorf representation. For the largest Fibonacci number that appears in the Zeckendorf representation, if it is the mth Fibonacci number and the original Fibonacci number is the nth Fibonacci number, then the following relationship between m and n will hold.

2 Examples

1~5: Itself

8(5th): 11. 11-8=3(3rd).

13: Itself

21(7th): 23. 23-21=2(2nd).

34(8th): 37. 37-34=3(3rd).

55(9th): 59. 59-55=4=3(3rd)+1

89: Itself

144(11th): 149. 149-144=5(4th).

233: Itself

377(13th): 379. 379-377=2(2nd).

3 Discussion

It seems to be related to the Lujandre conjecture because of the similarity in shape (1/2 power or 2 power).