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Topic: Fibonacci number starting with 123456789 (Read 705 times) 

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


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Grimbal
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Re: Fibonacci number starting with 123456789
« Reply #1 on: Aug 12^{th}, 2020, 6:38am » 
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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.


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towr
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Re: Fibonacci number starting with 123456789
« Reply #2 on: Aug 12^{th}, 2020, 10:55am » 
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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)

« Last Edit: Aug 12^{th}, 2020, 10:55am by towr » 
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Grimbal
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Re: Fibonacci number starting with 123456789
« Reply #3 on: Aug 13^{th}, 2020, 5:06pm » 
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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.

« Last Edit: Aug 13^{th}, 2020, 5:14pm by Grimbal » 
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towr
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Re: Fibonacci number starting with 123456789
« Reply #4 on: Aug 14^{th}, 2020, 2:28am » 
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I should have thought of that .5 Aside from that I used the same approach.

« Last Edit: Aug 14^{th}, 2020, 2:31am by towr » 
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