Depending on what you consider the length of 0 to be, there are either 6 or 7 one-digit Fibonacci numbers. There are 5 two-digit Fibonacci numbers. There are 4 four-digit Fibonacci numbers.
Prove that for n > 1, there are always either 4 or 5 n-digit Fibonacci numbers, or find a counterexample.
As Fibonacci numbers get larger, the ratio of F(i+1)/f(i) approaches the Golden Ratio Phi (1+sqrt(5))/2 = 1.61803... So the first n-digit Fibonacci will be somewhere between 100000xxxxx and 161803xxxxx.
Note that the base 10 log(phi) = 0.208987640249979 which is between 1/5 and 1/4, so multiplying the first n-digit Fibonacci by phi 3 times will never be enough to spill over into one more digit, and by phi 6 times will always produce a number one digit longer.
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