What non-zero Fibonacci numbers are one less than a power of two? (That would make each of them consist of all 1's in binary.)
I found this problem interesting and decided to do some digging and found a pdf. The title and authors:
Fibonacci numbers at most one away from a perfect power
Yann Bugeaud, Florian Luca,
Maurice Mignotte and Samir Siksek
They conclude the only Fibonacci perfect powers are 1, 8, and 144; the only Fibonacci numbers that are one less than a perfect power are 0, 3, and 8; the only Fibonacci numbers that are one more than a perfect power are 1, 2, and 5 (counting 0 as a perfect power).
This would then imply the answer to this puzzle are the obvious values 1 and 3.