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Fibonacci numbers represented as sum of integers

Source: Baltic Way 2017 Problem 3

November 14, 2017
algebraFibonacciAdditive Number TheoryExtremal combinatorics

Problem Statement

Positive integers x1,...,xmx_1,...,x_m (not necessarily distinct) are written on a blackboard. It is known that each of the numbers F1,...,F2018F_1,...,F_{2018} can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of mm? (Here F1,...,F2018F_1,...,F_{2018} are the first 20182018 Fibonacci numbers: F1=F2=1,Fk+1=Fk+Fk−1F_1=F_2=1, F_{k+1}=F_k+F_{k-1} for k>1k>1.)
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