TABLE OF RIGHT DIAGONALS

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

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 (205,²425,²565²) appear to have been obtained from 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.

It was shown previously that these numbers are a part of a sequence of squares and this page is a continuation of that effort.

I will show from scratch, (i.e. from first principles) that these tuples (a²,b²,c²) whose sum a² + b² + c² − 3b² = 0 are generated from another set of tuples that (except for the initial set of tuples) obeys the equation a² + b² + c² − 3b² ≠ 0.

Find the Initial Tuples

As was shown in the web page Generation of Right Diagonals, the first seven tuples of real squares, are generated using the formula c² = 2b² − 1 and placed into table T below. The first number in each tuple all a start with +1 which employ integer numbers as the initial entry in the diagonal.

The desired c² is calculated by searching all b numbers between 1 and 100,000. However, it was found that the ratio of b_n+1/b_n or c_n+1/c_n converges on (1 + √2)² as the b's or c's get larger. This means that moving down each row on the table each integer value takes on the previous b_n or c_n multiplied by (1 + √2)², i.e., 5.8284271247...

Furthermore, this table contains seven initial tuples in which all a start with −1 instead as of +1 as was shown in Part IA. This new route which differs slightly from Part IIB, and is easier to follow, may be employed and replace Part IIB. This will also apply to the other methods Part IIIB through Part VIB. The results, however, are identical, the same equations and magic squares are obtained by either method.

The initial simple tuple (−1,1,1) is the only tuple stands on its own. Our first example is then (−1,5,7).

Construction of two Tables of Right Diagonal Tuples

1. The object of this exercise is to generate a table with a set of tuples that obey the rule: a² + b² + c² − 3b² ≠ 0
   and convert these tuples into a second set of tuples that obey the rule: a² + b² + c² − 3b² = 0.
2. To generate table I we take the tuple (−1,5,7) and add 0 to each entry (except the −1) in the tuple to produce Table I with −1 entries in the first column.
3. We also set a condition for table I. We need to know two numbers e and g where g = 2e and which when added to the second and third numbers, respectively, in the tuple of table I produce the two numbers in the next row of table I.

Table I

−1	5	7
−1	5+e	7+g

4. These numbers, e and g are not initially known but a mathematical method will be shown below on how to obtain them. Having these numbers on hand we can then substitute them into the tuple equation 1² + (en + 5)² + (gn + 7)² − 3(en +5)² along with n (the order), the terms squared and summed to obtain a value S which when divided by a divisor d produces a number f.
5. This number f when added to the square of each member in the tuple (1,b,c) generates (f + 1)² + (f + en + 5)² + f + gn + 7)² − 3(f + en +5)² producing the resulting tuple in table II. This is the desired tuple obeying the rule a² + b² + c² − 3b² = 0.

Table I −1 5 7 −1 5+e 7+g

⇒

Table II −1 5 7 −1 + f 5+e + f 7+g + f

6. The Δs are calculated, the difference in Table 2 between columns 2 or 3, and the results placed in the last column.
7. Note that the third column in Table II is identical to column 2 but shifted up two rows.
8. The final tables produced after the algebra is performed are shown below:

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

Thus the values of the rows in both tables can be obtain by using a little arithmetic as was shown above or we can employ the two mathematical equations to generate each row. The advantage of using this latter method is that any n can be used. With the former method one calculation after another must be performed until the requisite n is desired.

1. Square A example of a magic square of order number n = 9 produced from the tuple (98, 122, 142). Squares B and C are magic squares of order n = 13 produced from the tuple (194, 226, 254). The magic sums in this case are 44652 and 153228, respectively.

This concludes Part IC. To continue to Part II.
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