Fibonacci Sums

Probability Level 2

aa, bb, and cc are distinct positive integers, and the sum of any two of these integers is a Fibonacci number.

What is the smallest possible value of a+b+ca+b+c?

If you think there are no possible values for aa, bb, and cc, please put the answer as 0.


Definition:

The first few Fibonacci numbers are 1,1,2,3,5,8,13,21,1,1,2,3,5,8,13,21, \ldots.

In general, they satisfy the following recursive relation: F(0)=F(1)=1,F(n)=F(n2)+F(n1)  for n>1.F(0) = F(1) = 1,\quad F(n) = F(n-2) + F(n-1) \ \text{ for } n > 1.

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