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Binary Repunit Fibonaccis (Posted on 2007-01-22) Difficulty: 5 of 5
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.)

See The Solution Submitted by Charlie    
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Research Results Comment 6 of 6 |
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.

  Posted by Brian Smith on 2016-07-02 20:10:42
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