The last digit of the numbers in the Fibonacci Sequence form a pattern that repeats after every 60th number:

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1

This pattern can be seen in the following list of the first 72 Fibonacci numbers:

0 | 0 |

1 | 1 |

2 | 1 |

3 | 2 |

4 | 3 |

5 | 5 |

6 | 8 |

7 | 13 |

8 | 21 |

9 | 34 |

10 | 55 |

11 | 89 |

12 | 144 |

13 | 233 |

14 | 377 |

15 | 610 |

16 | 987 |

17 | 1,597 |

18 | 2,584 |

19 | 4,181 |

20 | 6,765 |

21 | 10,946 |

22 | 17,711 |

23 | 28,657 |

24 | 46,368 |

25 | 75,025 |

26 | 121,393 |

27 | 196,418 |

28 | 317,811 |

29 | 514,229 |

30 | 832,040 |

31 | 1,346,269 |

32 | 2,178,309 |

33 | 3,524,578 |

34 | 5,702,887 |

35 | 9,227,465 |

36 | 14,930,352 |

37 | 24,157,817 |

38 | 39,088,169 |

39 | 63,245,986 |

40 | 102,334,155 |

41 | 165,580,141 |

42 | 267,914,296 |

43 | 433,494,437 |

44 | 701,408,733 |

45 | 1,134,903,170 |

46 | 1,836,311,903 |

47 | 2,971,215,073 |

48 | 4,807,526,976 |

49 | 7,778,742,049 |

50 | 12,586,269,025 |

51 | 20,365,011,074 |

52 | 32,951,280,099 |

53 | 53,316,291,173 |

54 | 86,267,571,272 |

55 | 139,583,862,445 |

56 | 225,851,433,717 |

57 | 365,435,296,162 |

58 | 591,286,729,879 |

59 | 956,722,026,041 |

60 | 1,548,008,755,920 |

61 | 2,504,730,781,961 |

62 | 4,052,739,537,881 |

63 | 6,557,470,319,842 |

64 | 10,610,209,857,723 |

65 | 17,167,680,177,565 |

66 | 27,777,890,035,288 |

67 | 44,945,570,212,853 |

68 | 72,723,460,248,141 |

69 | 117,669,030,460,994 |

70 | 190,392,490,709,135 |

71 | 308,061,521,170,129 |

72 | 498,454,011,879,264 |

Lucien Khan arranged these 60 digits of the pattern in a circle, as shown in illustration below:

Here he found other interesting results:

- The zeros align with the 4 cardinal points on a compass.
- The fives align with the 8 other points of the 12 points on a clock.
- Except for the zeros, the number directly opposite each number adds to 10.

Lucien postulates that ancient knowledge of these relationships contributed to the development of our modern use of 60 minutes in an hour, and presentation of numbers on the face of the clock.

I found too that any group of four numbers that are 90 degrees from each other (15 away from each other in the circle) sum to 20, except again for the zeros. As an example, use 1, 7, 9 and 3, which appear one to the right of each of the compass points.

Additionally, every group of five numbers that define the points of the 12 pentagons on the circle also create a pattern. Four of the pentagons have even-numbered last digits of 0, 2, 4, 6, and 8. The remaining eight pentagons have odd-numbered last digits of 1, 3, 5, 7 and 9.

Another interesting pattern yet was observed by Lucien Khan: The 216th number is this sequence is 619220451666590135228675387863297874269396512. The sum of all the digits in that number add up to 216, as well. He notes that it is believed that the secret or hidden name of God contains 216 characters. There are many other fascinating relationships and sacred geometries, which are presented by Lucien Khan in more detail at the links below.

**References:**

https://docs.google.com/document/d/1mVWd1aLiYZQU8VvYFBnW8kxodeYim3bYDIFfh-w42eU/pub

john shanahan says

The compass thing is very impressive but I think finding the name of God is going a bit too far!

Gary B Meisner says

Maybe, but if you do a Google search for “name of god 216 letters” you’ll find 1.6 million references. See https://www.google.com/search?q=name+of+god+216+letters

Roland says

“Maybe, but if you do a Google search for “name of god 216 letters” you’ll find 1.6 million references. ”

Only 1.6 mio? Well if it were really true it would have been 216 million. So, checkmate!

Gary B Meisner says

Not so fast there. First, your assumption that the two number should agree is not all that logical. Google is just a man made search engine, not a divine revelation of scripture or nature. Second, a checkmate is a check from which the king cannot escape. Just since my post, the number of Google references has increased to 2.1 million references. That means that human awareness is growing, but will never fully comprehend God. So, checkmate!

Sean says

Duly noted sir!

Peter Nockolds says

This makes an interesting connection with the dodecahedron which has 12 faces of 5 sides each. I wonder what would happen if we expressed the Fibonacci in sexuagesimal, base 60 notation as used by the Sumerians.

Gary B Meisner says

The use by the Sumerians of base 60 for their number scheme is added evidence that the number 60 had special significance to the ancients.

Jain 108 says

Just for the record, I wrote about this wheel of 60 in a published book in 2010, 3 years before Khan, showing the 60 Pattern and the cardinal alignment of the zeroes. It is in one of my 9 books from the series THE BOOK OF PHI, volume 3, sub-titled “The 108 Codes, an Introduction” pages 35 onwards, chapter 3, called Time Code 12:24:60 Encrypted in the Fibonacci Sequence.

To review or buy this book, click on the link

http://www.jainmathemagics.com/bookofphivol3/

and an ebook version is also available

http://www.jainmathemagics.com/ebookofphivol3/

Regards, Jain 108

G Cabilan says

I was surprised by the 216th fibonacci number find. Number of compound letters in the Tamil alphabet is 216. Interesting coincident isn’t it?

Marek says

I would guess this fact is probably the very reason why 216 is related to God’s name.

Since Tamil is the oldest culture and original language of the world, it makes sense that later cultures based on Tamil retains information of that great divine concept of alphabet but covered with the mist creating the mystery of secret God’s name consisting of 216 letters.

Btw, I am not Tamil. I am not saying that Tamil is oldest to promote myself. I am saying it because it is most probably the truth and it is unfortunately widely ignored, what in turn leads to mist and mystery instead of clear logical explanations.

Mark says

Older cultures didn’t have an alphabet, but correct me if I’m wrong. I am not sure what cultivation means, but I cannot imagin it started with an alphabet, let alone a number considdered to be a name.

David C. Irving says

In the example where the 60 numbers are arranged to define the five points of the twelve pentagons (five rows of twelve numbers). i found that by arranging the numbers in 12 rows of 5 numbers, 10 rows of 6 numbers, 6 rows of 10 numbers all produced the same interesting related patterns within each column. I began each arrangement starting with 1, as in the start of the sequence. ex. 1, 1, 2, 3, 5 etc. This subject of the Golden Ratio and the Fibonacci sequence and how it relates to God fascinates me and finding this information presented here has piqued my curiosity like never before. i’m hooked.

Sinisa Knezevic says

Do you have the visual presentation of your idea, or web site where we can read more about it.

Thanks!

Wojtek says

In this compas, right next to zeros there are always numbers 1,3,5,7 (typical endings of prime numbers) – this little detail can be a connection beetwen Fibonnaci numbers and Primes.

And if you check Lucas numbers you find a pattern of 12 digits that have an intersting connection to this compas too.

Mark says

I don’t know why it repeats with the power of 7 (1,7,9,3) one way and 3 (3,9,7,1) the other way, but the fact that those 2 mirror each other is rather far from a miracle.

mark says

Can be reduced to 20 numbers by doubling the sequence: 02246066280886404482

: D says

Fragment of a 3 x 20 matrix:

0 1 1 2 3 5 8

1 6 7 3 0 3 3

1 7 8 5 3 8 1

2 3 5 8 3 1 4

3 0 3 3 6 9 5

5 3 8 1 9 0 9

8 3 1 4 5 9 4

3 6 9 5 4 9 3

1 9 0 9 9 8 7

4 5 9 4 3 7 0

Mark says

Like this ?

11235

83145

94370

77415

61785

38190

99875

27965

16730

33695

49325

72910

Gary B Meisner says

Technically, the 0 you have in 72910 at the end should have been the first digit in 01123, which changes each of the 12 groups of 5 numbers you listed.

mark says

0 1 1 2 3

5 8 3 1 4

5 9 4 3 7

0 7 7 4 1

5 6 1 7 8

5 3 8 1 9

0 9 9 8 7

5 2 7 9 6

5 1 6 7 3

0 3 3 6 9

5 4 9 3 2

5 7 2 9 1

(thank you for the correction)

mark says

The Lucas-number equivalent:

2 1 3 4 7

1 8 9 7 6

3 9 2 1 3

4 7 1 8 9

7 6 3 9 2

1 3 4 7 1

8 9 7 6 3

9 2 1 3 4

7 1 8 9 7

6 3 9 2 1

3 4 7 1 8

9 7 6 3 9

KDub says

No zeros or fives

Mark says

Better to write it like this:

055055055055

189

13

21

34

55

89

Mark says

(the initial comment was meant as a reply to Wojtek February 2, 2018 at 8:06 am)

Mark says

Another nice line up:

1. 0 1. 1. 2. 3. 5. 8.

3. 1. 4. 5. 9. 4. 3

7. 0. 7. 7. 4. 1. 5. 6

1 7. 8. 5. 3. 8. 1

9. 0. 9. 9. 8. 7. 5. 2

7. 9. 6. 5. 1. 6. 7

3. 0. 3. 3. 6. 9 5. 4

9. 3. 2. 5. 7. 2. 9.

1 0. 1. 1. 2. 3. 5. 8

3. 1 ….

7 0. ..

1….

Mark says

That didn’t come across too well; it should have looked something like this:

1-0-1-1-2-3-5-8

-3-1-4-5-9-4-3

7-0-7-7-4-1-5-6

-1-7-8-5-3-8-1

etc.

Which can be read bottom/up (diagonally) as:

1-0-1-1-2-3-5-8

-7-0-7-7-4-1-5-6

1-9-0-9-9-8-7-5-2

-7-3-0-3-3-6-9-5-4

Joshua says

If you take every fifth number it produces the last digit cycle of the Lucas numbers. with exception of the zeros and fives.

213471897639

First digit is a zero so all we have is 055055055055

Next digit is 1897639 21347 Then it repeats 1897639 21347

Next digit is 13471897639 2 Then it repeats 13471897639 2

Next digit is 213471897639 Then it repeats 213471897639

Next digit is 3471897639 21 Then it repeats 3471897639 21

mark says

Astonishing the Lucas sequence is that present in the last digits of the Fibinacci sequence (apart from the binary string.)

In general it is quite strange both the decimal and the hexidecimal system suit the Fibonacci numbers that well, but maybe it’s the other way around which makes it even more mysterious.

Mark says

If you order the last digits of the Fibonacci-numbers in 5 rows of 12 as in the table above::

011235831459

437077415617

853819099875

279651673033

695493257291

each row can be turned into the same sequence of last Lucas numbers digIts by adding every second number (even and uneven n)

1347189763

1347189763

1347189763

1347189763

1347189763

I says

And if you take the natural numbers and multiply them by 4, the last digits are the first column

Mark says

The last digits of squared fibonacci numbers add up after a period of 15 numbers::

0, 1, 1, 4, 9, 25, 64, 169, 441, 1156, 3025, 7921, 20736, 54289, 142129, 372100,

974169, 2550409, 6677056, 17480761, 45765225, 119814916, 313679521, 821223649, 2149991424

The same accounts for golden rectangles::

0, 1, 2, 6, 15, 40, 104, 273, 714, 1870, 4895, 12816, 33552, 87841, 229970,

602070, 1576239, 4126648, 10803704,, 28284465, 74049690, 193864606, 507544127, 1328767776, 3478759200, 9107509825, 23843770274

Moshiya says

I’ve long known these series, but I’ll give you another,,not just the last digit but each number reduced to a single digit repeats itself after 24 times. So you got the minutes and here are the hours 😉

112358437189887641562819 then it starts again 1123584 etc. The reason and why it does so is to lengthily to explain here, but I can say this, there is structure behind numbers.

Moshiya says

This has long been known to me, you have the minutes so I will give the hours 24, and will repeat itself

Again take the whole numbers and reduce to single digit like 46368 is 4+6+3+6+8=27 2+7=9.

It would ne to much to write it all down hère but thé reason ans how van ne explained as thé is a structure upon which thèse numbers are placed..,this has far reaching consequences when shown so it can not be done

As a reply.

Mark says

All 60 numbers from a different perspective:

0 2 6 6 2 0 8 4 4 8 (0)

5 9 7 7 9 5 1 3 3 1 (5)

5 3 9 9 3 5 7 1 1 7 (5)

0 6 8 8 6 0 4 2 2 4 (0)

5 7 1 1 7 5 3 9 9 3 (5)

5 9 7 7 9 5 1 3 3 1 (5)

Extended:

0 2 6 6 2 0 8 4 4 8 0 2 6 6 2 0

5 9 7 7 9 5 1 3 3 1 5 9 7 7 9 5

5 3 9 9 3 5 7 1 1 7 5 3 9 9 3 5

0 6 8 8 6 0 4 2 2 4 0 6 8 8 6 0

5 7 1 1 7 5 3 9 9 3 5 7 1 1 7 5

5 9 7 7 9 5 1 3 3 1 5 9 7 7 9 5

0 8 4 4 8 0 2 6 6 2 0 8 4 4 8 0

5 1 3 3 1 5 9 7 7 9 5 1 3 3 1 5

5 7 1 1 7 5 3 9 9 3 5 7 1 1 7 5

0 4 2 2 4 0 6 8 8 6 0 4 2 2 4 0

5 3 9 9 3 5 7 1 1 7 5 3 9 9 3 5

5 1 3 3 1 5 9 7 7 9 5 1 3 3 1 5

0 2 6 6 2 0 8 4 4 8 0 2 6 6 2 0

5 9 7 7 9 5 1 3 3 1 5 9 7 7 9 5

5 3 9 9 3 5 7 1 1 7 5 3 9 9 3 5

0 6 8 8 6 0 4 2 2 4 0 6 8 8 6 0

5 7 1 1 7 5 3 9 9 3 5 7 1 1 7 5

5 9 7 7 9 5 1 3 3 1 5 9 7 7 9 5

M says

Arguably the most symmetrical representation of the last digits

0 3 9 4 3 5 2 1 1 2 5 3 4 9 3 0

0 1 3 8 1 5 4 7 7 4 5 1 8 3 1 0

0 7 1 6 7 5 8 9 9 8 5 7 6 1 7 0

0 9 7 2 9 5 6 3 3 6 5 9 2 7 9 0

mark says

This way it can be closed like a book with all 6o (55) digits adding up to 10 as in the circle

3 6 5 9 2 7 9 0 1 3 8 1 5 4 7

1 2 5 3 4 9 3 0 7 1 6 7 5 8 9

7 4 5 1 8 3 1 0 9 7 2 9 5 6 3

9 8 5 7 6 1 7 0 3 9 4 3 5 2 1

It can be viewed a product of these 2

3 8 5 3 2 1 1 0 1 1 2 3 5 8 3

3 8 5 3 2 1 1 0 1 1 2 3 5 8 3

3 8 5 3 2 1 1 0 1 1 2 3 5 8 3

3 8 5 3 2 1 1 0 1 1 2 3 5 8 3

1 7 9 3 1 7 9 0 1 3 9 7 1 3 9

7 9 3 1 7 9 3 0 7 1 3 9 7 1 3

9 3 1 7 9 3 1 0 9 7 1 3 9 7 1

3 1 7 9 3 1 7 0 3 9 4 1 3 9 7

: D says

This seems to be the underlying pattern (matrix) responsible for the repetition.

5 5 0 5 5 0 5 5 0 5 5

5 0 5 5 0 5 5 0 5 5 0

0 5 5 0 5 5 0 5 5 0 5

Since it also works with the natural numbers, it might have little to do with the Fibonacci-sequence and more with the decimal system.

0 1 2 3 4 5 6 7 8 9

5 6 2 8 9 5 1 2 8 4

0 6 7 3 9 0 6 2 3 9

0 1 2 3 4 5 6 7 8 9

(adding the binary matrix to the natural numbers)

Similar to the Liber Abaci-numbers

0 1 1 2 3 5 8 3 1 4 5

5 1 6 7 3 0 3 3 6 9 5

5 6 1 7 8 5 3 8 1 9 0

0 1 1 2 3 5 8 3 1 4 5

M says

In chronological order:

0 1 1 2 3 5 8 3 1 4 5 9 4 3 7 0 7 7 4 1

5 6 1 7 8 5 3 8 1 9 0 9 9 8 7 5 2 7 9 6

5 1 6 7 3 0 3 3 6 9 5 4 9 3 2 5 7 2 9 1

0 1 1 2 3 5 8 3 1 4 5 9 4 3 7 0 7 7 4 1

As the ‘clock’ turns with the power of three, so does each string when 5’s are subtracted.

0 1 1 2 3 0 3 3 6 9 0 9 9 8 7 0 7 7 4 1

This follows from a 3/5 sequence: 3,5,35,355,33555,33355555,3333355555555,

As every 5th is a 5-fold, another 055055-string is added:

0

1

1

2

3 0

0 5

3 5

3 0

6 5

9 5 0

0 0 5

9 5 5

9 5 0

8 0 5

7 5 5 0

0 5 0 5

7 0 5 5

7 5 5 0

4 5 0 5

1 0 5 5 0

0 5 5 0 5

.

Jain 108 says

I have made some new discoveries on the Fibonacci 60 Code, refer to my website:

https://www.jain108.com/fiboncci-60-code/

JAIN’S DISCOVERY: Fibonacci 60 Code: Spiral Pattern of ReEntry

I am proud to release another rare gem based on the Infinitely Repeating 60 Final Digits of the Fibonacci Sequence. When I plotted the 60 numbers on graph paper, (like the work of Ulam’s Rose revealing the 24 Pattern of Prime Numbers, and the stock-marketeering work of Gann’s Wheel of 24) I discovered that after 60 steps, the pattern RE-ENTERS ITSELF or ends where it begins, thus forming a predictive, cyclic pattern of the highest order that has applications with 60 hertz frequency and other hi-tech connections regarding frequency and vibration. It asks the question, did Tesla base his life-giving innovations on this distinctive 60 periodicity that obeys the Laws of Nature.

Jain 108

ps: I will explain this in further detail on Facebook soon. This revelation has never been published before in print and will appear as a 15 minute video discourse.

Gary B Meisner says

Hello Jain. Your page above at https://www.jain108.com/fiboncci-60-code/ is a real treasure of Fibonacci relationships, both mathematically and visually. Thanks much for your very creative and revealing work on this, Jain, and for sharing it with readers here. Regards, Gary Meisner

D says

Quite obvious, but yet not noted is the fact that one can read the circle counterclockwise as:

0, 1, -1, 2, -3, 5, -8 using subtraction instead of addition.

Mark says

There is another nice pattern based on Fibonacci squares.

The 72nd and last Fibonacci number in the list ends with the square of the sixth Fibonacci number (8) which is 64

72 = 2 x 6^2

Almost magically the 50th Fibonacci number ends with the square of the fifth Fibonacci number (5) because 50/2 is the square of 5.

So the square of the 4th Fibonacci number might correspond with the last digit(s) of the 2 x 4^2 = 2 x 16 = 32nd Fibonacci number; and yes it does. Here’s a little video showing (off) this relationship by forming a spiralling square of all the 60 last digits: https://youtu.be/-9IjAFBvPzg

Deja-vu says

Checking the end digits of the golden squares:

0, 1, 1, 4, 9, 5, 4, 9, 1, 6, 5, 1, 6, 9, 9, 0

0, 9, 9, 6, 1, 5, 6, 1, 9, 4, 5, 9, 4, 1, 1, 0

the repeating period is 30 digits long and has a cross-over decimal symmetry. that can be split in half once more revealing a familiar pattern.

0, 1, 1, 4, 9, 5,

5, 4, 9, 1, 6, 5,,

5, 1, 6, 9, 9, 0,

0, 9, 9, 6, 1, 5

5, 6, 1, 9, 4, 5,

5, 9, 4, 1, 1, 0,

Replier says

Each half can be split in 3 I meant, The last digit sequence of squared Lucas numbers 419691 has been encountered before in a 6 x 10 table of the Fibonacci last digit sequence:

011235831, 4

594370774, 1

561785381., 9

099875279, 6

516730336, 9

549325729, 1

(011235831, 4)

The nature of the relationship n –> 2n^2 is still a mystery to me; anyone ?

Mark D. says

Here’s a nice little arithmetic curiosity: 1/89=0.112358… (as known).

(But) 55/89 = 0,6179775280898876…which differs only in that it starts with the 21st last digit (times 10) and the 22nd. Moreover the inverse (89/55) is: 1,618181818181818

Gary B Meisner says

Yes, see more on 1/89 at https://www.goldennumber.net/89-and-109/.