The Diophantine Equation x2 + 3y2 = z2 (Part V)

A Method of Generating Triples from Novel Equations

This page is a method for generating triples of the form (x,y,z) for the Diophantine equation x2 + 3y2 = z2. The method is similar to a method employing imaginary numbers where the imaginary x is part of a magic square diagonal. In addition, the equations posted here differ in form from those in the magic square setting.

This new method is a way of producing triples from a set of novel equations that may be used to access the triples, x, y and z. The x, y and z as well as the equations are listed in the table headings below according to the following format:

δ1x y z δ2
a Da2 - n2 2na Da2 + n2

where a is a generating number starting at zero, the D is any integer greater than zero and the (n2, 2n) are values taken from table A:

Table A (n2 & 2n values)
n2149162536496481100...
2n2468101214161820...

Since there are an infinite number of (n2, 2n) integers there are also an infinite number of tables which can be tabulated with their accompanying equations. A second sequential access is also possible using the δ1 and δ2 columns whose δ1δ1 and δ2δ2 are both 6 for all the tables. In addition,the triples which are generated may be present in primitive or non primitive form, and these non primitives can be converted to primitive forms by dividing out any common factors.

Table I
δ1x y z δ2
a 3a2 - 1 2a 3a2 + 1
0-101
33
1224
99
211413
1515
326628
2121
447849
2727
5741076
3333
610712109
3939
714614148
Table II
δ1x y z δ2
a 3a2 - 4 4a 3a2 + 4
0-404
33
1-147
99
28816
1515
3231231
2121
4441652
2727
5712079
3333
610424112
3939
714328151

Now for Tables III and IV.

Table III
δ1x y z δ2
a 3a2 - 9 6a 3a2 + 9
0-909
33
1-6612
99
231221
1515
3181836
2121
4392457
2727
5663084
3333
69936117
3939
713842156
Table IV
δ1x y z δ2
a 3a2 - 16 8a 3a2 + 16
0-16016
33
1-13819
99
2-41628
1515
3112443
2121
4323264
2727
5594091
3333
69248124
3939
713156163

For Tables V and VI.

Table V
δ1x y z δ2
a 3a2 - 25 10a 3a2 + 25
0-25025
33
1-221028
99
2-132037
1515
323052
2121
4234073
2727
55050100
3333
68360133
3939
712270172
Table VI
δ1x y z δ2
a 3a2 - 36 12a 3a2 + 36
0-36036
33
1-331239
99
2-242448
1515
3-93663
2121
4124884
2727
53960111
3333
67272144
3939
711184183

Finally we have Tables VII and VIII.

Table VII
δ1x y z δ2
a 3a2 - 49 14a 3a2 + 49
0-49049
33
1-461452
99
2-372861
1515
3-224276
2121
4-15697
2727
52670124
3333
65984157
3939
79898196
Table VIII
δ1x y z δ2
a 3a2 - 64 16a 3a2 + 64
0-64064
33
1-611667
99
2-523276
1515
3-374891
2121
4-1664112
2727
51180139
3333
64496172
3939
783112211

Only eight triples, (x,y,z), were used in the calculation for the Diophantine equation x2 + 3y2 = z2 to get an idea of its versatility. Note that the tables produced can be expanded downward indefinitely. In addition, using the tables one can generate the appropriate equations that are useful for generating any triple in a table using a random access method or using δ1 and δ2 to access the (x,y,z)s sequentially.

This concludes Part V. Go to Part IV. Go to Part VI.

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Copyright © 2020 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com