### Post by mortlach on Sept 26, 2016 23:23:55 GMT

This will be a quick overview of

(7817 is the 988th prime.)

This means that every integer has a unique representation as a sum of Fibonacci numbers a ‘Fibonacci-Sum-Representation’ (FSR). This unique representation can also be expressed as the index (position) of the FSR in Fibonacci sequence, or their ‘Fibonacci-Sum-Index-Representation’ (FSIR). When we do this we are tacitly working in something known as the ‘Fibonacci Base’ number system (similar to binary, or decimal but instead of having powers of 2 or 10 we have Fibonacci numbers). Below is given example decimal , FSR, FSIR and The Fibonacci-Base representations of the first few integers:

Decimal / Fibonacci-Sum / Fibonacci-Sum-Index / Fibonacci-Base

0 {0} {1} 0

1 {1} {2} 1

2 {2} {3} 10

3 {3} {4} 100

4 {1, 3} {2, 4} 101

5 {5} {5} 1000

6 {1, 5} {2, 5} 1001

7 {2, 5} {3, 5} 1010

8 {8} {6} 10000

9 {1, 8} {2, 6} 10001

10 {2, 8} {3, 6} 10010

11 {3, 8} {4, 6} 10100

12 {1, 3, 8} {2, 4, 6} 10101

13 {13} {7} 100000

14 {1, 13} {2, 7} 100001

15 {2, 13} {3, 7} 100010

16 {3, 13} {4, 7} 100100

17 {1, 3, 13} {2, 4, 7} 100101

18 {5, 13} {5, 7} 101000

19 {1, 5, 13} {2, 5, 7} 101001

20 {2, 5, 13} {3, 5, 7} 101010

21 {21} {8} 1000000

We can use this table to go from a number in the decimal system to a number in the ‘Zeckendorf system.’ We can also do the reverse, mapping from the Fibonacci-Sum-Index and/or Fibonacci-Base representation to decimal. Consider the decimal representation of the following (carefully chosen) numbers:

{1} 0 0

{2} 1 1

{3} 10 2

{4} 100 3

{5} 1000 5

{6} 10000 8

{7} 100000 13

{8} 1000000 21

We are counting in ‘

A = 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987

Then take the prime that has the positional index for each A, with index 0 giving the first element (or simply take each A+1 ‘ th prime). These are:

B = 2, 3, 5, 7, 13, 23, 43, 79, 149, 263, 463, 829, 1481, 2593, 4507, 7817

Then take the Absolute value of | 3301 - B |

to give:

C = 3299, 3298, 3296, 3294, 3288, 3278, 3258, 3222, 3152, 3038, 2838, 2472, 1820, 708, 1206, 4516

If we arrange C in spiral (a form closely associated with the Fibonacci numbers) the square is reproduced.

*Comments, questions, suggestions, omissions etc ? please try #cicadasolvers

MSGA

*a way*to**exactly**reproduce the number square on Page 15. Some of the concepts are well known, like the prime and the Fibonacci sequence, some maybe not so well known like Zeckendorf's theorem and the Fibonacci-Base. The web has many resources on these items so we will only summarize here.**The Primes:**{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59…. 7817 }**The Fibonacci sequence:**{ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 }(7817 is the 988th prime.)

**Zeckendorf's theorem:**en.wikipedia.org/wiki/Zeckendorf%27s_theorem*Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. Zeckendorf's theorem has two parts:*

**Existence:**every positive integer n has a Zeckendorf representation.**Uniqueness:**no positive integer n has two different Zeckendorf representations.This means that every integer has a unique representation as a sum of Fibonacci numbers a ‘Fibonacci-Sum-Representation’ (FSR). This unique representation can also be expressed as the index (position) of the FSR in Fibonacci sequence, or their ‘Fibonacci-Sum-Index-Representation’ (FSIR). When we do this we are tacitly working in something known as the ‘Fibonacci Base’ number system (similar to binary, or decimal but instead of having powers of 2 or 10 we have Fibonacci numbers). Below is given example decimal , FSR, FSIR and The Fibonacci-Base representations of the first few integers:

Decimal / Fibonacci-Sum / Fibonacci-Sum-Index / Fibonacci-Base

0 {0} {1} 0

1 {1} {2} 1

2 {2} {3} 10

3 {3} {4} 100

4 {1, 3} {2, 4} 101

5 {5} {5} 1000

6 {1, 5} {2, 5} 1001

7 {2, 5} {3, 5} 1010

8 {8} {6} 10000

9 {1, 8} {2, 6} 10001

10 {2, 8} {3, 6} 10010

11 {3, 8} {4, 6} 10100

12 {1, 3, 8} {2, 4, 6} 10101

13 {13} {7} 100000

14 {1, 13} {2, 7} 100001

15 {2, 13} {3, 7} 100010

16 {3, 13} {4, 7} 100100

17 {1, 3, 13} {2, 4, 7} 100101

18 {5, 13} {5, 7} 101000

19 {1, 5, 13} {2, 5, 7} 101001

20 {2, 5, 13} {3, 5, 7} 101010

21 {21} {8} 1000000

We can use this table to go from a number in the decimal system to a number in the ‘Zeckendorf system.’ We can also do the reverse, mapping from the Fibonacci-Sum-Index and/or Fibonacci-Base representation to decimal. Consider the decimal representation of the following (carefully chosen) numbers:

{1} 0 0

{2} 1 1

{3} 10 2

{4} 100 3

{5} 1000 5

{6} 10000 8

{7} 100000 13

{8} 1000000 21

We are counting in ‘

*powers of Fibonacci*’ numbers. With the above we can now recreate the ‘Number Square’. Find the decimal representation of the first 16 powers of the*‘Fibonacci Base’*ie. 0, 1, 10, 100, 1000 to 10^16 in the ‘Fibonacci Base’:A = 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987

Then take the prime that has the positional index for each A, with index 0 giving the first element (or simply take each A+1 ‘ th prime). These are:

B = 2, 3, 5, 7, 13, 23, 43, 79, 149, 263, 463, 829, 1481, 2593, 4507, 7817

Then take the Absolute value of | 3301 - B |

to give:

C = 3299, 3298, 3296, 3294, 3288, 3278, 3258, 3222, 3152, 3038, 2838, 2472, 1820, 708, 1206, 4516

If we arrange C in spiral (a form closely associated with the Fibonacci numbers) the square is reproduced.

*Comments, questions, suggestions, omissions etc ? please try #cicadasolvers

MSGA