Dear all,
This article expresses some amusing findings between prime numbers, Fibonacci numbers, and golden ratio.
First of all, golden ratio can be achieved by the ratio of two CONSECUTIVE Fibonacci numbers. As you know, golden ratio = 1.61803 = 610/377 = 987/610 etc.
Next, let us look at the first 1,000 digits of Golden Ratio as below
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454749246185695364864449241044320771344947049565846788509874339442212544877066478091588460749988712400765217057517978834166256249407589069704000281210427621771117778053153171410117046665991466979873176135600670874807101317952368942752194843530567830022878569978297783478458782289110976250030269615617002504643382437764861028383126833037242926752631165339247316711121158818638513316203840052221657912866752946549068113171599343235973494985090409476213222981017261070596116456299098162905552085247903524060201727997471753427775927786256194320827505131218156285512224809394712341451702237358057727861600868838295230459264787801788992199027077690389532196819861514378031499741106926088674296226757560523172777520353613
I have conducted the same analysis like what I have done with PI. In detail, this series can be fragmented into CONSECUTIVE prime or Fibonacci numbers.
First of all, we have 1618033988749. This is a prime. Then we have 89, a Fibonacci number
Next to it is 4848204586834365638117720309179, which is also a prime. Similarity, the next value is 8, a Fibonacci number, and so on etc.
You can view my detail calculation in file golden-1.docx which has been shared for everyone in my google drive
https://docs.google.com/document/d/1Z15QpBxVm0x3r2mfOQJB5HTmKgNGjPaZCoIlKa-buqk/edit
My investigation can be used for further researching with more digits of golden ratio, for example 2000, 3000, or 100,000 digits.
All of your feedback are welcome.
It seems that there is some link between sequences (e.g. Fibonacci) and quasicrystals
https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/
Great, thank you Alexander Kushnerov , I will read your article carefully to see if any experiment can be done
Dear Thinh,
Nice observations.
All the mentioned numbers are approximations for the golden ratio.
The exact value of the golden ratio can be determined as lim Fn+1/Fn as n goes to infinity equals (1+sqrt(5))/2.
So nobody can confirm your observations as n goes to infinity.
Notice that your observations may be taken into consideration if and only if you can discover a periodic pattern and not an arbitrary one.
Wish you good luck to catch some pattern if any, that helps to go with the infinite expansion Of (1+sqrt(5))/2.
Best regards
Something tells me that one should not be surprised to find an ability to construct arbitrary groupings upon lists of symbols (in this case, digits) such that another loosely defined rule should regularly, even consistently, apply. If the condition were some kind of regular progression (which seems to be missing in the given demonstration), then I might be more inclined to view the results as significant; there seems no regularity within the results that captures the imagination.
Cj Nev ,
Could you please explain how your answer is related to the question?
Especially: " elliptical or shortest path the speed of light follows. "!!
Issam, sure. Briefly, to explain and relate my answer to the question, prime distribution and Fibonacci sequences, if not all sequences, approach the maximal limit of the golden ratio. Primes in particular are in one-to-one correspondence with an ellipse that light also travels, beginning relatively linearly (constant acceleration c; undifferentiated photons) equidistant along the ellipse (major axis) as primes 1, 2, 3, before then doubling the gap to twin primes 5 and 7 at the sharper end of the ellipse (minor axis; c-squared, or orthogonal electromagnetism; fractionally-differentiated light), there doubling the gap again to 11 and 13, and so on. Please see this application for I hope further clarity:
ELLIPTIC CURVES AND PRIMALITY PROVING
A. O. L. ATKIN
F. MORAIN
December 23, 1992
Cj Nev ,
Your answer is still not related to the current question. It is not associated with any scientific fact! There is no logical flow, just words besides words. The mentioned paper
"ELLIPTIC CURVES AND PRIMALITY PROVING
A. O. L. ATKIN,F. MORAIN,December 23, 1992"
of Atkin and Morain is related to primality testing and very far from the context of your previous post!! I hope you will show serious answers.
In https://doi.org/10.1016/j.jnt.2017.06.014
We find the following :
(1) Fn ≡ 1 (mod n) if and only if one of the following four congruences holds:
i) n ≡ 1 (mod 20),
ii) n ≡ 9 (mod 20),
iii) n ≡ 11 (mod 20),
iv) n ≡ 19 (mod 20).
(2) Fn ≡ -1 (mod n) if and only if one of the following four congruences holds:
i) n ≡ 3 (mod 20),
ii) n ≡ 7 (mod 20),
iii) n ≡ 13 (mod 20),
iv) n ≡ 17 (mod 20).
where Fn is the nth Fibonacci number.
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