A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

Picture of a square

Production of New Tables

This page continues from the previous Part IC. The purpose is to expand the first two tuples of Table XI (even number 288 and Table XII (odd number 289) and show that the tuples (±i,408,577) generates a different b for each table as well as different number of subtuples generated from the expansion of both tuples.

We take the first four lines of each table as shown below:

Table XI (Even Number 288)
δ1iai b cδ2
-288i0288
289i289
i408577
Table XII (Odd Number 289)
δ1iai b cδ2
-289i0289
288i288
-i408577

In order to keep the tables small enough we expand these two section of both tables to produce the following subtables where the even number n is incremented/decremented using the odd numbers:

Table XIex (Even Number 288)
δ1iai b cδ2
-288i0288
1i1
-287i24289
3i3
-284i48292
5i5
-279i72297
7i7
-272i96304
9i9
-263i120313
11i11
-252i144324
13i13
-239i168337
15i15
-224i192352
17i17
-207i216369
Table XIex (Even Number 288)
δ1iai b cδ2
19i19
-188i240388
21i21
-167i264409
23i23
-144i288432
25i25
-119i312457
27i27
-92i336484
29i29
-63i360513
31i31
-32i384544
33i33
i408577

and the odd number n is incremented/decremented using even numbers:

Table XIIex (Odd Number 289)
δ1iai b cδ2
-289i0289
2i2
-287i34291
6i6
-281i68297
10i10
-271i102307
14i14
-257i136321
18i18
-239i170339
22i22
-217i204361
Table XIIex (Odd Number 289)
δ1iai b cδ2
26i26
-191i238387
30i30
-161i272417
34i34
-127i306451
38i38
-89i340489
42i42
-47i374531
6i46
-i408577

The final tuple for each is either (+i,408,577) or (-i,408,577) both of which lead to (-1, 166464, 332929) upon squaring. Note that the central number in the tuple, b, is incremented by 24 in the first case and 34 in the second. In addition, all the tuples generate legitimate right diagonals upon squaring even the ones that are factorable.

This concludes Part IIC. Go to either Part IIIC or Part ID to continue on tables of allowed tuples.

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Copyright © 2016 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com