

Title: Fibonacci number starting with 123456789 Post by towr on May 13^{th}, 2020, 12:59pm Can you find a Fibonacci number that starts with 123456789... ? 

Title: Re: Fibonacci number starting with 123456789 Post by Grimbal on Aug 12^{th}, 2020, 6:38am I can't give it in full, but I can give you the start of it. :) More seriously, given that Fibonacci numbers are based on the golden ratio, the initial digits should cover quite uniformly all possibilities. You should find every sequence of 9 digits among the initial digits of the first few bilions of Fibonacci numbers. But since the sequence is exponential these numbers tend to be huge, especially near the end. 

Title: Re: Fibonacci number starting with 123456789 Post by towr on Aug 12^{th}, 2020, 10:55am There's a way to find one reasonably quickly. (i.e. a lot faster than generating each Fibonacci number in turn and check.) And basically the question is: what is lowest N such that the Nth fibonacci number starts with 123456789 Another fun question is to find the first Fibonacci number with more than 7 (or 8) digits so that the first 7 (or 8) and last 7 (or 8) digits are the same (or mirror each other) 

Title: Re: Fibonacci number starting with 123456789 Post by Grimbal on Aug 13^{th}, 2020, 5:06pm I would propose F_{158349400}. https://www.wolframalpha.com/input/?i=Fibonacci+158349400 It has 33093068 digits. You understand I can't give it in full. I wrote a clever program that solves phi^{n}/sqrt(5) ~= 123456789.5 * 10^{k} by solving n log(phi)  k log(10) ~= log(123456789.5 * sqrt(5)) for integer n and k. But it found a solution with k<0 and n<0. Trying to limit k and n to positive values it fails due to the limited precision of doubles in java. So in the end I wrote a brute force program that gave me the answer in a few seconds. 

Title: Re: Fibonacci number starting with 123456789 Post by towr on Aug 14^{th}, 2020, 2:28am I should have thought of that .5 Aside from that I used the same approach. 

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