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        riddles >> medium >> fibonacci iteration
        
(Message started by: Christine on Mar 26th, 2014, 10:12am)
Title: fibonacci iteration
Post by Christine on Mar 26th, 2014, 10:12am
If you take any Fibonacci number (except 1, 13)
find the sum of the squares of its digits, and then repeat; you will come to 89

why is that?
Title: Re: fibonacci iteration
Post by towr on Mar 26th, 2014, 10:39am
It's true for 856929 of the first million non-negative integers, so it's not special to Fibonacci numbers.

Also, it's not true for quite a few Fibonacci numbers, e.g.:
17 :  2584
18 :  4181
24 :  75025
31 :  2178309
32 :  3524578
44 :  1134903170
49 :  12586269025
82 :  99194853094755497
83 :  160500643816367088
86 :  679891637638612258
92 :  12200160415121876738
105 :  6356306993006846248183
108 :  26925748508234281076009
114 :  483162952612010163284885
116 :  1264937032042997393488322
121 :  14028366653498915298923761
125 :  96151855463018422468774568
129 :  659034621587630041982498215
131 :  1725375039079340637797070384
132 :  2791715456571051233611642553
133 :  4517090495650391871408712937
142 :  343358302784187294870275058337

In fact, it's about the same fraction as for regular integers, again pointing to Fibonacci numbers not being special.
Title: Re: fibonacci iteration
Post by Christine on Mar 26th, 2014, 11:10am
Thanks.
I've just looked at prime curios! page
http://primes.utm.edu/curios/page.php/89.html

One entry, it reads:
Hugo Steinhaus (1887-1972) proved that if you take any positive integer, find the sum of the squares of its digits, and then repeat; that you will either come to 1 or the cyclic sequence 145, 42, 20, 4, 16, 37, 58, 89.

It doesn't mention about percentage of getting a 89 or any percentages of these numbers
Title: Re: fibonacci iteration
Post by rmsgrey on Mar 27th, 2014, 5:11am
http://en.wikipedia.org/wiki/Happy_numbers seems relevant
Title: Re: fibonacci iteration
Post by Christine on Mar 30th, 2014, 1:23pm
thanks


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