Jebreel-magic Square Among Fibonacci

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Smarandache Magic Square among Fibonacci & Lucas Numbers By: Muneer Jebreel Karama SS-Math-Hebron Field Education Officer Jerusalem .box 19149 [email protected] Abstract The aim of this paper is to present new Smarandache magic squares related to Fibonacci and Lucas numbers Key words Magic square, Fibonacci number, Lucas number, . Smarandache Magic Square among Fibonacci & Lucas Numbers with degree n. Introduction: The Fibonacci numbers satisfy (Reference: 1).

L

n+ 2

=

L

F

n +1

n+ 2

+

=

L

F and the Lucas numbers L F + F , F =0 , F =1 n

n

n+1

n

,

n

L

0

=2 ,

0

1

L =1 1

.

A magic square is the arrangement of the numbers from 1 to n2 in an n × n matrix with each number occurring exactly once (Reference: 2), so the number in each row , column, and main diagonal is supposed to sum to the same number. Definition: The Smarandache magic square of Fibonacci and Lucas with degree n , is composed of four magic squares. Two of them Fibonacci numbers and the other two Lucas numbers (see figure.1 below):

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2Fn+5

9Fn+5

4Fn+5

7Fn+5

5Fn+5

3Fn+5

6Fn+5

Fn+5

8Fn+5

4 Fn

3Fn

8Fn

6Ln+2

7Ln+2

2Ln+2

9Fn

5Fn

Fn

Ln+2

5Ln+2

9Ln+2

2Fn

7Fn

6Fn

8Ln+2

3Ln+2

4Ln+2

8Ln+3

Ln+3

6Ln+3

3Ln+3

5Ln+3

7Ln+3

4Ln+3

9Ln+3

2Ln+3

Figure. 1 Smarandache magic square of Fibonacci and Lucas number with degree n. If we let n = 4 (without loss of generality) we obtain the following Smarandache magic square of Fibonacci and Lucas number with degree 4(i.e. n =4) as in figure.2

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68

306

136

238

170

102

204

34

272

12

9

24

108

126

36

27

15

3

18

90

162

6

21

18

144

54

72

232

29

174

87

145

203

116

261

58

Figure. 2 Smarandache magic square of Fibonacci and Lucas number with degree 4 .

Figure .1 includes the following interesting properties: 1) The sum of the squares of Fibonacci, and the sum of the squares of Lucas of the corresponding cells are equal. Example:

F

2 n +5

+

F

2 n

=

L

2 n +3

+

L

2 n +2

.

So if n = 4 then we have figure.2 such that: 902 + 1452 = 152 + 1702 2) The square of the sum of any two or more cells, or any rows, columns, or diagonals in Fibonacci squares is equal to the squares of the sum of corresponding two or more cells, rows, columns, or diagonals of Lucas squares.

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Example:

(8 L

(8 F + F n

+ 6 F n ) + (8 F n+ 5 + n 2

F

+ 6 F n+ 5) = n+ 5 2

+L n +3 +6 L n +3 ) + (8 Ln +2 + Ln +2 + 6 Ln +2 ) 2

n +3

2

3) The square of the total of all cells in Fibonacci squares equals to the squares of the total of all the cells in Lucas squares. Example:

(F

(L

+ 2 F n + ... + 9 F n ) + n 2

(F

+ 2 F n + 5 + ... + 9 F n + 5 ) = n+ 5 2

+ 2 Ln +3 +... +9 Ln+3 ) + ( Ln+2 + 2 Ln +2 + ... + 9 Ln+2 ) n +3 2

2

4) Each matrix is a magic square, so the sum of all the rows, columns, and diagonals are identical within the square. 5) By substituting n=1, 2, 3… , we can construct infinite Smarandache Fibonacci and Lucas magic squares. Smarandache Concatenated Magic Squares of Fibonacci and Lucas It is formed from the concatenation of numbers in magic squares such as the sum of the numbers in any horizontal, vertical, line is always the same constant, and the main diagonal is the same constant if the concatenation between two magic squares or more , but not in the same magic square.

1)

It is important to say that; There are infinitely many Smarandache Concatenated Magic Squares formed from any magic square.

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2) There are many types of concatenation , so we can concatenating inside any magic square and the diagonals are not the same constant ( see [3] for more examples ) , but concatenating from one to the other, the diagonals will remain the same constant such as the following example :

06810 8

30612 6

13603 6

23801 8

17009 0

10216 2

20414 4

03405 4

27207 2

01206 8

00930 6

02413 6

10823 2

12602 9

03617 4

02723 8

01517 0

00310 2

01808 7

09014 5

16220 3

00620 4

02103 4

01827 2

14411 6

05426 1

07205 8

23201 2

02900 9

17402 4

08702 7

14501 5

20300 3

11600 6

26102 1

05801 8

Open Question : Are there a Smarandache Concatenated Magic Squares of Fibonacci and Lucas with prime numbers ?

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Reference 1) http://math.holycross.edu/~davids/fibonacci/fibonacci.htl 2) Adler, Allan. What is Magic Square? The Math Form .Drexel University. 3 April http://mathform.org/alejandre/magic.square/adler.whatsqua re.html. 3) M.Jebreel Karama , Smarandache Concatenated Magic Squares , Smarandache Notion Journal ,2004,14: 80- 83.