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:
| δ1 | x |
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)
n2 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | ... |
2n | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | ... |
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
| δ1 | x |
y | z |
δ2 |
a | |
3a2 - 1 |
2a |
3a2 + 1 | |
0 | | -1 | 0 | 1 | |
| 3 | | | | 3 |
1 | | 2 | 2 | 4 | |
| 9 | | | | 9 |
2 | | 11 | 4 | 13 | |
| 15 | | | | 15 |
3 | | 26 | 6 | 28 | |
| 21 | | | | 21 |
4 | | 47 | 8 | 49 | |
| 27 | | | | 27 |
5 | | 74 | 10 | 76 | |
| 33 | | | | 33 |
6 | | 107 | 12 | 109 | |
| 39 | | | | 39 |
7 | | 146 | 14 | 148 | |
|
|
Table II
| δ1 | x |
y | z |
δ2 |
a | |
3a2 - 4 |
4a |
3a2 + 4 | |
0 | | -4 | 0 | 4 | |
| 3 | | | | 3 |
1 | | -1 | 4 | 7 | |
| 9 | | | | 9 |
2 | | 8 | 8 | 16 | |
| 15 | | | | 15 |
3 | | 23 | 12 | 31 | |
| 21 | | | | 21 |
4 | | 44 | 16 | 52 | |
| 27 | | | | 27 |
5 | | 71 | 20 | 79 | |
| 33 | | | | 33 |
6 | | 104 | 24 | 112 | |
| 39 | | | | 39 |
7 | | 143 | 28 | 151 | |
|
Now for Tables III and IV.
Table III
| δ1 | x |
y | z |
δ2 |
a | |
3a2 - 9 |
6a |
3a2 + 9 | |
0 | | -9 | 0 | 9 | |
| 3 | | | | 3 |
1 | | -6 | 6 | 12 | |
| 9 | | | | 9 |
2 | | 3 | 12 | 21 | |
| 15 | | | | 15 |
3 | | 18 | 18 | 36 | |
| 21 | | | | 21 |
4 | | 39 | 24 | 57 | |
| 27 | | | | 27 |
5 | | 66 | 30 | 84 | |
| 33 | | | | 33 |
6 | | 99 | 36 | 117 | |
| 39 | | | | 39 |
7 | | 138 | 42 | 156 | |
|
|
Table IV
| δ1 | x |
y | z |
δ2 |
a | |
3a2 - 16 |
8a |
3a2 + 16 | |
0 | | -16 | 0 | 16 | |
| 3 | | | | 3 |
1 | | -13 | 8 | 19 | |
| 9 | | | | 9 |
2 | | -4 | 16 | 28 | |
| 15 | | | | 15 |
3 | | 11 | 24 | 43 | |
| 21 | | | | 21 |
4 | | 32 | 32 | 64 | |
| 27 | | | | 27 |
5 | | 59 | 40 | 91 | |
| 33 | | | | 33 |
6 | | 92 | 48 | 124 | |
| 39 | | | | 39 |
7 | | 131 | 56 | 163 | |
|
For Tables V and VI.
Table V
| δ1 | x |
y | z |
δ2 |
a | |
3a2 - 25 |
10a |
3a2 + 25 | |
0 | | -25 | 0 | 25 | |
| 3 | | | | 3 |
1 | | -22 | 10 | 28 | |
| 9 | | | | 9 |
2 | | -13 | 20 | 37 | |
| 15 | | | | 15 |
3 | | 2 | 30 | 52 | |
| 21 | | | | 21 |
4 | | 23 | 40 | 73 | |
| 27 | | | | 27 |
5 | | 50 | 50 | 100 | |
| 33 | | | | 33 |
6 | | 83 | 60 | 133 | |
| 39 | | | | 39 |
7 | | 122 | 70 | 172 | |
|
|
Table VI
| δ1 | x |
y | z |
δ2 |
a | |
3a2 - 36 |
12a |
3a2 + 36 | |
0 | | -36 | 0 | 36 | |
| 3 | | | | 3 |
1 | | -33 | 12 | 39 | |
| 9 | | | | 9 |
2 | | -24 | 24 | 48 | |
| 15 | | | | 15 |
3 | | -9 | 36 | 63 | |
| 21 | | | | 21 |
4 | | 12 | 48 | 84 | |
| 27 | | | | 27 |
5 | | 39 | 60 | 111 | |
| 33 | | | | 33 |
6 | | 72 | 72 | 144 | |
| 39 | | | | 39 |
7 | | 111 | 84 | 183 | |
|
Finally we have Tables VII and VIII.
Table VII
| δ1 | x |
y | z |
δ2 |
a | |
3a2 - 49 |
14a |
3a2 + 49 | |
0 | | -49 | 0 | 49 | |
| 3 | | | | 3 |
1 | | -46 | 14 | 52 | |
| 9 | | | | 9 |
2 | | -37 | 28 | 61 | |
| 15 | | | | 15 |
3 | | -22 | 42 | 76 | |
| 21 | | | | 21 |
4 | | -1 | 56 | 97 | |
| 27 | | | | 27 |
5 | | 26 | 70 | 124 | |
| 33 | | | | 33 |
6 | | 59 | 84 | 157 | |
| 39 | | | | 39 |
7 | | 98 | 98 | 196 | |
|
|
Table VIII
| δ1 | x |
y | z |
δ2 |
a | |
3a2 - 64 |
16a |
3a2 + 64 | |
0 | | -64 | 0 | 64 | |
| 3 | | | | 3 |
1 | | -61 | 16 | 67 | |
| 9 | | | | 9 |
2 | | -52 | 32 | 76 | |
| 15 | | | | 15 |
3 | | -37 | 48 | 91 | |
| 21 | | | | 21 |
4 | | -16 | 64 | 112 | |
| 27 | | | | 27 |
5 | | 11 | 80 | 139 | |
| 33 | | | | 33 |
6 | | 44 | 96 | 172 | |
| 39 | | | | 39 |
7 | | 83 | 112 | 211 | |
|
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.
Go back to homepage.
Copyright © 2020 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com