Jun 2003
Suva, Fiji
2^{3}×3×5×17 Posts

Fibonacci modulo Fibonacci
Just playing around with Fibonaccis and I notice, at least for small Fibonaccis, that there are integers 6,9,14,15,16,17,19 which never appear as mod values for Fibonaccis mod (any Fibonacci).
For example: F(11)=89 is mod 1,2,4,1,11,5,21,34,0,89,89..... the first 13 Fibonacci numbers, 89 repeating thereafter.
Looking at it the other way around, the first 44 Fibonaccis 1,1,2,3,5,8,13,21,34,55,89,144,233.... are 1,2,3,4,5,8,13,21,34,55,0,55,55,21,76,8,84,3,87,1,88,0,88,88,87,86,84,81,76,68,55,34,0,34,34,68,13,81,5,86,2,88,1,0 mod 89 and then the pattern repeats for the next 44 Fibonaccis.
Interestingly, the series 6,9,14,15,16,17,19.. does not appear in OEIS and therefore it is unclear to me if this property has been investigated.
Anyway, I would proposed based on extremely limited observations that such integers exist and further, there are infinite integers that are never mod values.
Maybe some mathematically minded person can (i) point me to the name of this property, and its proof or disproof, or (ii) prove or disprove either of these propositions.
