EXPERIENCING THE PURPLE ZEBRA (Fibonacci factorizations)

I went looking online for a list of the numbers in the Fibonacci series because I had discovered that every 8th value in that series is evenly divisible by 7. I wanted more of the series so I could test that further. Not only did I find more values for the Fibonacci series (http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibtable.html), but I found that someone had factorized the values up to the 1000-th value (http://mersennus.net/fibonacci/f1000.txt).

What I found is likely not a new discovery since it seems evident that whoever calculated the list of factors for the 1000 values and typed it up, has already noticed the factors have a way of repeating.

These repetitions are mind-boggling. This is like finding a pink and purple stripped zebra. Or a green sky over pink mountains. How can this be? Maybe there is some logical explanation. Maybe not.

Here are some of the repeating factors. Starting with factor-5 because it is the most startling (a “^” means “to the power of”):

5

Factor of 5 occurs one every 5 values

Factor of 5^2 occurs one every 25 values (5 x 5)

Factor of 5^3 occurs one every 125 values (5 x 25) or (5 x 5 x 5)

Factor of 5^4 occurs one every 625 values (5 x 125) or (5 x 5 x 5 x 5)

Notice that the intervals between 5-factors and powers thereof vary according to factors of 5!

3

Factor of 3 occurs one every 4 values

Factor of 3^2 occurs one every 12 values (3 x 4)

Factor of 3^3 occurs one every 36 values (3 x 12) or (3 x 3 x 4)

Factor of 3^4 occurs one every 108 values (3 x 36) or (3 x 3 x 3 x 4)

Factor of 3^5 occurs one every 324 values (3 x 108) or (3 x 3 x 3 x 3 x 4)

Notice that the intervals between the powers of 3-factors vary according to factors of 3!

7

Factor of 7 occurs one every 8 values

Factor of 7^2 occurs one every 56 values (7 x 8)

Factor of 7^3 occurs one every 392 values (7 x 56) or (7 x 7 x 8)

Notice that the intervals between the powers of 7-factors vary according to factors of 7!

11

Factor of 11 occurs one every 10 values

11^2 occurs one every 110 values (11 x 10)

13

Factor of 13 occurs one every 7 values

Factor of 13^2 occurs one every 91 values (13 x 7)

17

Factor of 17 occurs one every 9 values

Factor of 17^2 occurs one every 153 values (17 x 9)

19

Factor of 19 occurs one every 18 values

Factor of 19^2 occurs one every 342 values (19 x 18)

23

Factor of 23 occurs one every 24 values

Factor of 23^2 occurs at value 552 (23 x 24)

29

Factor of 29 occurs one every 14 values

Factor of 29^2 occurs one every 406 values (29 x 14)

31

Factor of 31 occurs one every 30 values

Factor of 31^2 occurs one every 930 values (31 x 30)

How can the frequency of the square consistently be the factor itself times the interval at which it occurs?

37

Factor of 37 occurs one every 19 values

Factor of 37^2 occurs at value 703 (37 x 19)

41

Factor of 41 occurs one every 20 values

Factor of 41^2 occurs at value 820 (41 x 20)

43

Factor of 43 occurs one every 44 values

47

Factor of 47 occurs one every 16 values

Factor of 47^2 occurs at value 752 (47 x 16)

53

Factor of 53 occurs one every 27 values

59

Factor of 59 occurs one every 58 values

61

Factor of 61 occurs one every 15 values

Factor of 61^2 occurs at value 915 (61 x 15)

2 – a bit different

Factor of 2 occurs one every 3 values

There is no 2^2 factor

Factor 2^3 occurs one every 6 values (2 x 3)

Factor 2^ alternates with 2

Pattern is 2, 2^3, 2, 2^4, 2, 2^3, 2, 2^5* repeat

*The value of 2^5 alternates with 2^6, then 2^7, then 2^6, then 2^8, then 2^6, then 2^7

Have you noticed that of the prime factors I’ve investigated here, many have the interval of repetition at a value that is + 1 or – 1 relative to the factor. That is: 2 at intervals of 3; 3 at intervals of 4; 7 at intervals of 8; 11 at intervals of 10; 19 at intervals of 18; 23 at intervals of 24; 31 at intervals of 30; 43 at intervals of 44; 59 at intervals of 58. And it continues with 67 at intervals of 68; 79 at 78; 83 at 84; 127 at 128; 131 at 130; 163 at 164; 179 at 178; 223 at 224; 239 at 238; 251 at 250; 283 at 284. What a feast for those searching for surprises!

Caveat: what I have included here in this post is only a sampling of the factors, and of course I have not analyzed the entire Fibonacci series which is infinite.

10

Factor of 2 x 5 occurs one every 15 values starting with the first value of 0

For Bible fans: Intersection of factor of 10 and factor of 7^2 (complex factor of 70 x 7) happens at value #840 (15 x 56)

Notice that 840 is also 12 x 70, both special numbers in the Bible.

Could the biblical “70 x 7” be an enigmatic reference to the interval between occurrences of factor 7^3 in the Fibonacci number series, which is every 392 values or 7 x 56? “70 x 7” in base 8 (octal number system) equates to “56 x 7” in base 10 (our decimal number system). Yes, I know that’s a stretch (since every time 70 x 7 is revealed by Sower’s numbers the calculations are base 10), but I had to mention it. What could be more awesome mathematically than these Fibonacci numbers? And more worthy of mention in the Bible? But I don’t have any examples of the ancients using base 8.

(Neat base calculator at http://www.cleavebooks.co.uk/scol/calnumba.htm)

Q. If I find all this intriguing, does that mean that my mind and the mind of whoever or whatever *designed* these repetitions have something in common? A. Of course there could be some mathematical reason for the repetitions of which I am clueless and in fact, no designer whatsoever. If there is such a Designer, It is light-years ahead of me, that’s for sure.

Until I know the answer, I’ll just be satisfied with a process of discovery.

First published November 23, 2014

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