TABLE OF RIGHT DIAGONALS

GENERATION OF RIGHT DIAGONALS FOR MAGIC SQUARE OF COMPLEX SQUARES (Part IIIA)

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 (97²,113²,127²) appears to have come out of the blue. 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.

Generation of Diagonal Tuple Sets

As was shown in the web page Generation of Right Diagonals, the first seven tuples of complex squares, i.e., those with real and imaginary coefficients are generated using the formula c² = 2b² + 1 and placed into table T_i below. The first number in each tuple of all a start with either +i or −i which employ imaginary 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...

The initial simple tuple (±i,1,1) is the only tuple stands on its own. Our third example is then (±i, 70, 99).

Construction of 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. The initial row, however, of table I is identical to the first row of table II. On the other hand, there are other examples where this is not true.
2. To accomplish this we set a condition. 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. Every first number, however, remains an i.

Table I

±i	70	99
±i	70+e	99+g

3. 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 (±i)² + (en + 70)² + (gn + 99)² − 3(en +70)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.
4. This number f when added to the square of each member in the tuple (1,b,c) generates (f ±i)² + (f + en + 70)² + f + gn + 99)² − 3(f + en +70)² producing the resulting tuple in table II. This is the desired tuple obeying the rule a² + b² + c² − 3b² = 0.

Table I ±i 70 99 ±i 70+e 99+g

⇒

Table II ±i 70 99 ±i + f 70+e + f 99+g + f

5. This explains why when both n and f are both equal to 0 that the first row of both tables are equal.
6. Two Tables II and III are set up. The first takes the values in Table I adds the requisite value from table f to generate Table II. These complex numbers are then squared to generate Table III, which gives the diagonals to be used per magic square.
7. The Δs are calculated, by subtracting column 2 from column 3 in Table III Squared and the results placed in the last column. For example adding 57 + 24i to −9 + 40i gives 48 + 64i and adding 57 + 24i to that value gives 105 + 88i.
8. The final tables produced after the algebra is performed are shown below. The divisor d is easily obtained using the algebraic equations below. The value for d = 82 ± 2i and f is found using equations (g), (h) and finally (i).

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. The following complex magic squares are exhibited first in square format and in square root format where both forms are equivalent. Only five complex squares with integer values for both the real and imaginary part were found for each of the squares looked at. For Square A (n = 1) the right diagonal tuple is (164 ± 5i, 292 ± 4i, 379 ± 4i) while for Square B (n = 2) the right diagonal tuple is (492 ± 13i, 678 ± 12i, 823 ± 12i) Unsquared. The magic sums in this case are 255744 ± 7008i and 1378620 ± 48816i, respectively.
2. Note that the top + and − go together and the and the bottom − and + also go together.

Magic square A Squared 146784 ± 119720i −34665 ∓ 115744i 143625 ± 3032i 82089 ∓ 114352i 85248 ± 2336i 88407 ± 119024i 26871 ± 1640i 205161 ± 120416i 23712 ∓ 115048i

≡

Magic square A Unsquared (410 ± 146i)2 70263 ± 14832i (379 ± 4i)2 102113 ± 18192i (292 ± 4i)2 (344 ± 173i)2 (164 ± 5i)2 16785 ± 5328i 86188 ± 16512i

Magic square B Squared 269620 ± 1243248i 431875 ∓1214184i 677185 ± 19752i 867105 ∓ 1207224i) 459540 ± 16272i 51975 ± 1239768i 241895 ± 12792i 487265 ± 1246728i 649460 ∓ 1210704i

≡

Magic square B Unsquared (878 ± 708i)2 431875 ∓1214184i (823 ± 12i)2 867105 ∓ 1207224i (678 ± 12i)2 (804 ± 771i)2 (492 ± 13i)2 487265 ± 1246728i 649460 ∓ 1210704i

3. We can get six complex squares from n = 2 by using squares that look similar but which are not. For example, (492 + 13i)² and (−13 + 492i)² are different numbers.

Magic square C Squared −459540 ∓ 16272i 1160975 ± 45336i 677185 ± 19752i 1596265 ± 52296i) 459540 ± 16272i −677185 ∓ 19752i 241895 ± 12792i −241895 ∓ 12792i 1378620 ∓ 48816i

≡

Magic square C Unsquared (−12 ± 678i)2 1160975 ± 45336i (823 ± 12i)2 1596265 ± 52296i (678 ± 12i)2 (−12 ∓ 823i)2 (492 ± 13i)2 (−13 ± 492i)2 1378620 ∓ 48816i

4. The following equations were used for calculating squares and square roots:
   Multiplication = (a + bi) × (a + bi) = a² − b² + 2abi
   Division = (a + bi) / (a + bi)
   Square root of (a + bi) :
   r = sqrt(a² + b²)
   r1 = x + y
   r2 = −x − y
   where,
   y = sqrt((r − a) / 2) and x = b / 2y

This concludes Part IIIA.
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