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   Fibonacci number starting with 123456789
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   Author  Topic: Fibonacci number starting with 123456789  (Read 446 times)
towr
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Fibonacci number starting with 123456789  
« on: May 13th, 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 12th, 2020, 6:38am »
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I can't give it in full, but I can give you the start of it.  Smiley
 
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 12th, 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 12th, 2020, 10:55am by towr » IP Logged

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Grimbal
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Re: Fibonacci number starting with 123456789  
« Reply #3 on: Aug 13th, 2020, 5:06pm »
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I would propose F158349400.
 
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
phin/sqrt(5) ~= 123456789.5 * 10k
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 13th, 2020, 5:14pm by Grimbal » IP Logged
towr
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Re: Fibonacci number starting with 123456789  
« Reply #4 on: Aug 14th, 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 14th, 2020, 2:31am by towr » IP Logged

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