A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES
THE USE OF ONE IMAGINARY NUMBER AS PART OF THE RIGHT DIAGONAL (Part IIIC)
Production of New Tables
This page continues from the previous Part IC. The next two tables employing numbers 1681 and 1682,
respectively, are Tables XIII, on the left, and XIV, on the right, and the Sum of each tuple, i.e., every other line is:
S = a^{2}
+ b^{2} + c^{2}
shown at the extreme right and are both identical.
Again we start off with either of these two numbers and fill up the tables by adding either 1682 to 1681 or 1681 to 1682 (with or without the
is). The δs are incremented by 3364 for Table XIII and 3362 for Table XIV.
The δs are then added to the previous a or c. The b is calculated as previously
according to equation:
[(2a/n + 2)^{1/2} × n]
where n is in our first case either 1681 or 1682.
Analysis of these tables shows that only table XIV contains tuples which are divisible by 2 (in light blue) to generate Table XIV_{subset 1}.
Table XIII (Odd Number 1681)
δ_{1}i  ai 
b  c  δ_{2} 
 1681i  0  1681  
1682i     1682 
 i  2378  3363  
5046i     5046 
 5047i  4756  8409  
8410i     8410 
 13457i  7134  16819  
11774i     11774 
 25231i  9512  28593  
15138i     15138 
 40369i  11890  43731  
18502i     18502 
 58871i  14268  62233  
21866i     21866 
 80737i  16646  84099  



Table XIV (Even Number 1682)
δ_{1}i  ai 
b  c  δ_{2} 
 1682i  0  1682  
1681i     1681 
 i  2378  3363  
5043i     5043 
 5042i 
4756  8406  
8405i     8405 
 13447i  7134  16811  
11767i     11767 
 25214i 
9512  28578  
15129i     15129 
 40343i  11890  43707  
18491i     18491 
 58834i 
14268  62198  
21853i     21853 
 80687i  16646  84051  

 
XI or XII
Sum 
0 

16998307 

67858608 

152681868 

271434432 

424116300 

610727472 

831267948 

The subtable Table XIV_{subset 1} below with odd number 841
is expanded by adding δ = 2 to ±841
to generate the first tuple. From this tuple we may calculate b_{initial}, followed by
division of b_{final} by b_{initial} to afford 162, the number of tuples that are required to fill in
the expanded table. Consequently, using the Gauss equation:
Sum = ½[n(x_{initial} + x_{final})]
which we rewrite to conform to our values as:
Sum of δs = ½[b_{final}/b_{initial}
(δ_{initial} + δ_{final})]
and entering in the values
3362 = ½[41(2 + δ_{final})]
δ_{final} = 162
Thus the table is composed of a Δ_{ca} of 1682,
a b_{initial} = 58 and a b_{final} = 2378. Thus there are 41 δs starting at
2(i) and ending at 162(i) of which only seven are shown.
[Note that n(i) is my shorthand version to stand for n or n(i)]
Table XIV_{subset 1} (Odd Number 841)
δ_{1}i  ai 
b  c  δ_{2} 
 841i  0  841  
3362i     3362 
 2521i  2378  4203  
10086i     867 
 12607i  4756  14289  
11760i     11760 
 29417i  7134  31099  

 
Table XIV_{subset 2} (Odd Number 841)
δ_{1}i  ai 
b  c  δ_{2} 
 841i  0  841  
2i     2 
 839i  58  843  
6i     6 
 833i  116  849  
10i     10 
 823i  174  859  
14i     14 
 809i  238  873  
18i     18 
 791i  290  891  
22i     22 
 769i  348  913  
...  ...  ...  ...  ... 
162i     162 
 2521i  2378  3363  

A second switcheroo places Table XV with an even number on the left and Table XVI with an odd number on the right. The
Δ_{ca} of the leftmost even table is 19600 while that of the rightmost and a second part of the odd table
below it is 19602.The center table corresponds to the magic sum and is identical in both cases.
The table is split so that the portion from 9801 to 342999 is used for
comparison with table XV. The extra lines (after division by 9), however, affords four more yellow lines to be used in Table XVI_{subset 1}.
Furthermore, the light green tuples of Table XV were divided by 4 to afford Table XV_{subset 1}.
Table XV (Even Number 9800)
δ_{1}i  ai 
b  c  δ_{2} 
 9800i  0  9800  
9801i     9801 
 i  13860  19601  
29403i     29403 
 29404i 
27720  49004  
49005i     49005 
 78409i  41580  98009  
68607i     68607 
 147016i 
55440  166616  
88209i     88209 
 235225i  69300  254825  
107811i     107811 
 343036i 
83160  362636  
127413i     127413 
 470449i  97020  490049  

 
XV or XVI
Sum 
0 

576298800 

2305195200 

5186689200 

9220780800 

14407470000 

20746756800 

28238641200 

 
Table XVI (Odd Number 9801)
δ_{1}i  ai 
b  c  δ_{2} 
 9801i  0 
9801  
9800i     9800 
 i  13860  19601  
29400i     29400 
 29399i  27720  49001  
49000i     49000 
 78399i 
41580  98001  
68600i     68600 
 146999i  55440  166601  
88200i     88200 
 235199i  69300  254801  
107800i     107800 
 342999i 
83160  362601  
127400i     127400 
 470399i  97020  490001  

Table XVI (Odd Number 9801) cont'd
147000i     147000 
 617399i  110880  637001  
166600i     166600 
 783999i 
124740  803601  
A portion of both tables XV and XVI can expanded further to give Table XV_{subset 1} and Table XVI_{subset 1}, with
even number 2450 and odd number 1089, respectively.
Expansion of Table XV_{subset 1} from just 2450 to 2351 affords Table XV_{subset 2} with
a Δ_{ca} of 4900, a b_{final}/b_{initial} = 99 and final
δs of 197i and 197, respectively, by the Gauss theorem.
Table XV_{subset 1} (Even Number 2450)
δ_{1}i  ai 
b  c  δ_{2} 
 2450i  0  2450  
9801i     9801 
 2351i  6930  12251  
29403i     29403 
 36754i  13860  41654  
49005i     49005 
 85759i  20790  90659  

  
Table XV_{subset 2} (Even Number 2450)
δ_{1}i  ai 
b  c  δ_{2} 
 2450i  0  2450  
1i     1 
 2449i  70  2451  
3i     3 
 2446i  140  2454  
...  ...  ...  ...  ... 
197i     197 
 2351i  6930  12251  

Similarly expansion of Table XVI_{subset 1} from just 1089 to 8711 affords Table XVI_{subset 2} with
a Δ_{ca} of 2178, a b_{final}/b_{initial} = 70 and final
δs of 278i and 278, respectively, by the Gauss theorem.
Table XVI_{subset 1} (Odd Number 1089)
δ_{1}i  ai 
b  c  δ_{2} 
 1089i  0  1089  
9800i     9800 
 8711i  4620  10889  
29400i     29400 
 38111i  9240  40289  
49000i     49000 
 87111i  13860  89289  

 
Table XVI_{subset 2} (Odd Number 1089)
δ_{1}i  ai 
b  c  δ_{2} 
 1089i  0  1089  
2i     2 
 1087i  66  1091  
6i     6 
 1081i  132  1097  
...  ...  ...  ...  ... 
278i     278 
 8711i  4620  10889  

This concludes Part IIIC. Go to Part ID to continue on tables of allowed tuples.
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Copyright © 2016 by Eddie N Gutierrez. EMail: enaguti1949@gmail.com