Sunday, 5 August 2012

Fibonacci Digit Sums

David Brooks sent me a note recently with the following interesting tidbits on the sum of the digits of some Fibonacci numbers.  Thought it was fun, so here they are.  Thanks David:

btw:  here is  OEIS sequence  
Find the next one, and send it here...
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 I ran across this interesting information while researching information on Leon Bankoff.

The sum of the digits of the 5th Fibonacci number is 5.  (Leon Bankoff) (Ok, the F(5) is five, so that one is a gimme'
The sum of the digits of the 10th Fibonacci number (55) is 10. (Leon Bankoff)
The sum of the digits of the 31st Fibonacci number is 31 (1346269).  (Leon Bankoff)
The sum of the digits of the 35th Fibonacci number is 35.  (Leon Bankoff)
The sum of the digits of the 62nd Fibonacci number is 62.  (Leon Bankoff)
The sum of the digits of the 72nd Fibonacci number is 72.  (Leon Bankoff)
The sum of the digits of the 175th Fibonacci number is 175.  (OEIS)
The sum of the digits of the 180th Fibonacci number is 180.  (OEIS)
The sum of the digits of the 216th Fibonacci number is 216.  (OEIS)
The sum of the digits of the 251st Fibonacci number is 251.  (OEIS)
The sum of the digits of the 252nd Fibonacci number is 252.  (OEIS)
The sum of the digits of the 360th Fibonacci number is 360.  (OEIS)
The sum of the digits of the 494th Fibonacci number is 494.  (OEIS)
The sum of the digits of the 504th Fibonacci number is 504.  (OEIS)
The sum of the digits of the 540th Fibonacci number is 540.  (OEIS)
The sum of the digits of the 946th Fibonacci number is 946.  (OEIS)
The sum of the digits of the 1188th Fibonacci number is 1188.  (OEIS)
The sum of the digits of the 2222nd Fibonacci number is 2222.  (OEIS)
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More mature students should read Joshua Zucker's comment below to begin to understand a little more about the frequency with which these occur.

Younger students might think about the digital root of the Fibonacci numbers since that only requires the sum of two simple one digit numbers. 


They might want to explore questions like:
How frequently is the digital root of F(n) the same as the digital root of n (obviously in all the case above, but are there others?)


There be a point where the sequence of digital roots repeats; Why?
 How would that information help you find large numbers for which the digital root of F(n) is the same as the digital root of n? 
Can you see how this would provide a shortcut in the search for numbers in like the sequence David Brooks has written about above?
What is the longest possible string length of one digit numbers before a repeating sequence must emerge?  (Perhaps you might attempt that question with only two digits, zero and one, or three digits and then expand the question).

There are some other interesting things to explore, for instance the sum of the digits of  F(2314) is 2134.   
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