A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

THE USE OF ONE IMAGINARY NUMBER AS PART OF THE RIGHT DIAGONAL (Part IE)

Production of New Tables

This page continues from the previous Part ID. The next series of tables are Tables XXI and XXII, employing numbers 1940449 and 1940450, respectively. Again we start off with either of these two numbers and fill up the tables by adding either 1940449 to 1940450 or 1940450 to 1940449. The δs are incremented by 3880900 for Table XXI and 3880898 for Table XXII. The δs are then added to the previous a or c and the b is calculated according to equation:

where n is either ±1940449 or ±1940450.

In addition, the following equation calculates for the sum of each tuple, i.e.:

which is the sum of the right major diagonal of a magic square and identical for both tables. Previously these sums have been listed in a separate table. However, the sum of the second lines are now over a billion and a steadily increasing. What is important are that the sums generated by both tables are identical, period.

Table XXII_{subset 1} is expanded by adding δ = 2 to ±970225 to generate the first tuple as shown in Table XXII_{subset 2}. From this tuple we may calculate b_initial, followed by division of b_final by b_initial to afford 3363, the number of tuples that are required to fill in the expanded table. Consequently, using the Gauss equation:

which we rewrite to conform to our values as:

and entering in the values

Thus, the table is composed of a Δ_c-a of 1940450, a b_initial = 1970 and a b_final = 2744210. Thus there are 1393 δs starting at 2(i) and ending at 5570(i) of which only three are shown. [Note n(i) is my shorthand version to stand for n or ni.]

Next we perform a switcheroo with Tables XXIII and XXIV, placing the even number 11309768 on the left and the odd number 11309769 on the right. We start by adding either 11309768 to 11309769 as in Table XXIII or 11309769 to 11309768 as in Table XXIV. The δs are incremented by 22619538 for Table XXIII and 2261936 for Table XXIV. The δs are then added to the previous a or c and the b is calculated according to equation:

where n is either ±11309768 or ±11309769. Again the sum of both tuples from each table is identical.

The light orange tuples of Table XXIII, whose numbers are all even, are factorable by 4, as shown in Table XXIII_{subset 1}. The Δ_c-a, however, is now 5654884 compared to 22619536 for those tuples of Table XXIII.

Table XXIII_{subset 1} is expanded by adding δ = 1 to ±2827442 to generate the first tuple as shown in Table XXIII_{subset 2}. From this tuple we may calculate b_initial, followed by division of b_final by b_initial to afford 3363, the number of tuples that are required to fill in the expanded table. Consequently, using the Gauss equation from above:

Thus, the table is composed of a Δ_c-a of 5654884, a b_initial = 2378 and a b_final = 7997214. Thus there are 3363 δs starting at 1(i) and ending at 6725(i) of which only three are shown. [Note n(i) is my shorthand version to stand for n or ni.]

This concludes Part IE. Go to Part IF to continue on tables of allowed tuples.

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