TABLE OF RIGHT DIAGONALS GENERAL METHOD

GENERATION OF RIGHT DIAGONALS FOR MAGIC SQUARE OF SQUARES (Part IA)

Square of Squares Tables

Andrew Bremner's article on squares of squares included the 3x3 square:

The numbers in the right diagonal as the tuple (205²,425²,565²) seem to have been derived elsewhere.. But I will show that this sequence is part of a larger set of tuples having the same property, i.e. the first number in the tuple when added to a difference (Δ) gives the second square in the tuple and when this same (Δ) is added to the second square produces a third square. All these tuple sequences can be used as entries into the right diagonal of a magic square.

We will show a general method for generating the squares for a right diagonal of a magic square. Beginning with the the tuple (1,b1, c1) we can generate the tuple (a, b, c) which when squared gives the diagonal numbers. Initially either b1 or c1 will be equal to ± k where k is any natural number 1,2,3,4.... Again the end result is that a₁² + b₁² + c₁² − 3b₁² ≠ 0 = S is transformed into a² + b² + c² − 3b² = 0 a necessary condition for the square to be magic.

Several examples will be shown, where k initially takes on the value ±3. The four sections are Part IA where c = −3, located on this page, Part IIA where c = 3, Part IIIA where b = −3 and Part IVA where b = 3.

To summarize the tuples of Table II below will be used as entries into a right diagonal of a magic square. Knowing the difference (b² − a²) or (c² − b²) will give us a value Δ which can be used to produce other entries into the magic square. To date only one magic square containing 7 entries has been found. Most other squares will contain 6 entries.

As to the reason for the picture of a square, the entries to the square occur as three tuples,viz, (a,b,c), (l,m,n) and (x,y,z) showing their connectivity. In addition, six or more of these entries are present as their squares.

Generation of Tables where c₁ = -3

1. The object of this exercise is to generate a Table I with a set of tuples that obey the rule: a₁² + b₁² + c₁² − 3b₁² ≠ 0 and convert these tuples into a second set of tuples (Table II) that obey the rule: a² + b² + c² − 3b² = 0.
2. Only k = 2j + 1 will be used, since the use of even k's generates half integer numbers.
3. In addition, we need to know two numbers e and g where g = 2e which when added to the b₁ and c₁ numbers of Table I, produce the next line of numbers (n + 1) in the next row of Table I. The number a₁ will always be 1, while e may vary depending on b or c.
4. Two other numbers f and d are calculated using the equation
   f = [2e²n² + (4c₁ − 4b₁) en +(1 − 2b₁² + c₁²)] / {2(2b₁ − c₁ − 1)}
   where n is the line number of the tables. f can also be generated directly from Table II from S/d. However, the value of d is equal to the denominator of the general equation above.
5. Finally Δs are calculated by taking the difference in Table II between (b² − a²) or (c² − b²), and the results placed under the Δ column. Both differences must be the same.
6. As an example we begin with the tuple (1,1,−3), where a₁ = 1, b₁ = 1 and c₁ = −3 and use the equation to generate f.
   f = [2e²n² + (−12 −4)en + 8]/2×4 = [2e²n² − 16en + 8]/8
   Setting e = 2 and g = 4 affords f = n² −4n + 1
   Substituting for f in (b) gives
   
   a = (n² −4n + 1 + 1 ) = (n² −4n + 2 )
   b = (n² −4n + 1 + 2n + 1) = (n² −2n + 2)
   c = (n² −4n + 1 + 4n − 3) = (n² − 2)
   
7. Substituting the appropriate n into the equations for a, b, and c produces Table II below. Using a computer program and the requisite calculations produced the tables below. As can be seen taking the value of f from the middle table and adding to a₁, b₁, c₁, produced a, b, c, respectively of Table II.

To obtain e, g, f, and d the algebraic calculations are performed as follows:

1. The first is a non magic square of squares listed in Bremner's paper is not magic but has been corrected to be in A with the magic sum (S_m) 38307. Two other examples are B and C produced from the tuple (23, 37, 47) and (98, 122, 142) as their squares. The magic sum, S_m, for these cases are 4107 and 44652, respectively, while the n's are 7 and 12, respectively.

This concludes Part IA. To continue to Part IB which treats tuples of the type (1,1,−13).
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