## Pattern | Fibonacci Num3ers- sum of square of digits is not 89

The first few Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, …

0^2   =   0
1^2   =   1
1^2   =   1

We also reach the number 89 with the next numbers in the series:
21,   34,   55,   89,   144,   233,   377,   610,   987,   1597

We don’t reach 89 with these pairs and triplet :

There are many Fibonacci numbers that don’t reach 89.
It’s interesting to find the peculiar ones, such as
pairs, triplets, quadruplets of consecutive Fibonacci numbers that don’t reach 89.

Check out:

Unhappy Number
http://mathworld.wolfram.com/UnhappyNumber.html

Happy Number
http://mathworld.wolfram.com/HappyNumber.html

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## About benvitalis

math grad - Interest: Number theory
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### 6 Responses to Pattern | Fibonacci Num3ers- sum of square of digits is not 89

1. Paul says:

I’m reckoning there is some proof lurking in there, but on a brute force method I looked up to (Fn100000), some 20899 digits and those were the only 2 numbers to be found, what was surprising was even at that number of digits there wasn’t one that went over 12 iterations.
considering that 987 has 10 iterations. (Fn100000) terminates after 7 iterations.
Paul.

• benvitalis says:

these pairs of consecutive Fibonacci numbers do not reach 89:
18 2584
19 4181

32 2178309
33 3524578

83 99194853094755497
84 160500643816367088

132 1725375039079340637797070384
133 2791715456571051233611642553

• benvitalis says:

Check out the update on the blog

2. Paul says:

When I mentioned the only 2 numbers I meant 1 and 89 as they were the 2 numbers mentioned in the examples shown. I didn’t look for pairs of or triples of the number 1 but I did notice quite a few. I will take another look and look for some runs.

Paul.