MathSciDoc: An Archive for Mathematician ∫

This paper investigates the sum of all non-Fibonacci integers that lie strictly between consecutive Fibonacci numbers. Starting from the operation a(n) = Mn− Fn, where Mn is that sum and Fn the preceding Fibonacci number, we obtain a sequence whose first three terms are themselves Fibonacci numbers: 1, 8, 34. We extend the exploration to seven other arithmetic operations (subtracting the next Fibonacci, adding either, multiplying either, and dividing either), and we systematically search for Fibonacci numbers that appear in each resulting sequence. We term such Fibonacci entries “lucky Fibonacci numbers”. Among all variants, only the addition and subtraction operations yield any lucky Fibonacci numbers: 1, 8, 34 (subtract previous), 5 (subtract next, ignoring the negative initial term), 21, 55 (add next), and none for add previous, nor for multiplication or division. We then apply the same gap-summing idea to other recurrent sequences: Lucas, Pell, and Tribonacci. For each, we find a small set of lucky Fibonacci numbers. All conjectures are verified computationally up to n = 200 (and up to Fibonacci F400), with no further hits found. The growth rates are exponential, making further coincidences increasingly improbable.

Fibonacci sequence, integer gaps, derived sequences, arithmetic operations, lucky Fibonacci numbers, conjectures in sequences.