The Diophantine Equation x2 − Dy2 = z2 (Part II)
A Method of Generating Triples from Novel Equations
A new, general method for generating triples of the form
(x,y,z) for the Diophantine equation x2 − Dy2 = z2, where D, the coefficient of
y can be any integer greater than zero is being introduced here.
The method produces a set of novel second order equations which generates the triples by either random or sequential means.
These equations are listed below using the following table heading format:
| δ1 | x |
y | z |
δ2 |
k | |
Dk2 + n
| mk |
Dk2 - n | |
where k is a counter starting at zero, D is the coefficient of y and (n, m) are values taken from the table below so that whenever
n = j2 then
m = 2j for all j > 0:
Table of n & m values
n | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | ... |
m | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | ... |
Since there are an infinite number of (n, m) integers there are also an infinite number of tables which can be tabulated with their accompanying equations; equations whose purpose is to generate the triples either by random or sequential access. A second sequential access is also possible using the δ1 and
δ2 columns whose δ1δ1
and δ2δ2 are both 2 for all the tables. In addition, a substantial number of triples are composed of numbers which can be simplified further via prime division. When D equals one the Diophantine equation takes the form of
x2 − y2 = z2. The calculations for this equation are shown in Tables I thru VIII. Note tha z may be negative at times, the reason being that the square root of
z2 can take on the values ±z as is verified by the
δ2 differences.
Table I
| δ1 | x |
y | z |
δ2 |
k | |
k2 + 1 |
2k |
k2 - 1 | |
0 | | 1 | 0 | -1 | |
| 1 | | | | 1 |
1 | | 2 | 2 | 0 | |
| 3 | | | | 3 |
2 | | 5 | 4 | 3 | |
| 5 | | | | 5 |
3 | | 10 | 6 | 8 | |
| 7 | | | | 7 |
4 | | 17 | 8 | 15 | |
| 9 | | | | 9 |
5 | | 26 | 10 | 24 | |
| 11 | | | | 11 |
6 | | 37 | 12 | 25 | |
| 13 | | | | 13 |
7 | | 50 | 14 | 48 | |
|
|
Table II
| δ1 | x |
y | z |
δ2 |
k | |
k2 + 4 |
4k |
k2 - 4 | |
0 | | 4 | 0 | -4 | |
| 1 | | | | 1 |
1 | | 5 | 4 | -3 | |
| 3 | | | | 3 |
2 | | 8 | 8 | 0 | |
| 5 | | | | 5 |
3 | | 13 | 12 | 5 | |
| 7 | | | | 7 |
4 | | 20 | 16 | 12 | |
| 9 | | | | 9 |
5 | | 29 | 20 | 21 | |
| 11 | | | | 11 |
6 | | 40 | 24 | 32 | |
| 13 | | | | 13 |
7 | | 53 | 28 | 45 | |
|
Next comes Tables III and IV.
Table III
| δ1 | x |
y | z |
δ2 |
k | |
k2 + 9 |
6k |
k2 - 9 | |
0 | | 9 | 0 | -9 | |
| 1 | | | | 1 |
1 | | 10 | 6 | -8 | |
| 3 | | | | 3 |
2 | | 13 | 12 | -5 | |
| 5 | | | | 5 |
3 | | 18 | 18 | 0 | |
| 7 | | | | 7 |
4 | | 25 | 24 | 7 | |
| 9 | | | | 9 |
5 | | 34 | 30 | 16 | |
| 11 | | | | 11 |
6 | | 45 | 36 | 27 | |
| 13 | | | | 13 |
7 | | 58 | 42 | 40 | |
|
|
Table IV
| δ1 | x |
y | z |
δ2 |
k | |
k2 + 16 |
8k |
k2 - 16 | |
0 | | 16 | 0 | -16 | |
| 1 | | | | 1 |
1 | | 17 | 8 | -15 | |
| 3 | | | | 3 |
2 | | 20 | 16 | -12 | |
| 5 | | | | 5 |
3 | | 25 | 24 | -7 | |
| 7 | | | | 7 |
4 | | 32 | 32 | 0 | |
| 9 | | | | 9 |
5 | | 41 | 40 | 9 | |
| 11 | | | | 11 |
6 | | 52 | 48 | 20 | |
| 13 | | | | 13 |
7 | | 65 | 56 | 33 | |
|
Next comes Tables V and VI.
Table V
| δ1 | x |
y | z |
δ2 |
k | |
k2 + 25 |
10k |
k2 - 25 | |
0 | | 25 | 0 | -25 | |
| 1 | | | | 1 |
1 | | 26 | 10 | -24 | |
| 3 | | | | 3 |
2 | | 29 | 20 | -21 | |
| 5 | | | | 5 |
3 | | 34 | 30 | -16 | |
| 7 | | | | 7 |
4 | | 41 | 40 | -9 | |
| 9 | | | | 9 |
5 | | 50 | 50 | 0 | |
| 11 | | | | 11 |
6 | | 61 | 60 | 11 | |
| 13 | | | | 13 |
7 | | 74 | 70 | 24 | |
|
|
Table VI
| δ1 | x |
y | z |
δ2 |
k | |
k2 + 36 |
12k |
k2 - 36 | |
0 | | 36 | 0 | -36 | |
| 1 | | | | 1 |
1 | | 37 | 12 | -35 | |
| 3 | | | | 3 |
2 | | 40 | 24 | -32 | |
| 5 | | | | 5 |
3 | | 45 | 36 | -27 | |
| 7 | | | | 7 |
4 | | 52 | 48 | -20 | |
| 9 | | | | 9 |
5 | | 61 | 60 | -11 | |
| 11 | | | | 11 |
6 | | 72 | 72 | 0 | |
| 13 | | | | 13 |
7 | | 85 | 84 | 13 | |
|
Finally we have Tables VII and VIII.
Table VII
| δ1 | x |
y | z |
δ2 |
k | |
k2 + 49 |
14k |
k2 - 49 | |
0 | | 49 | 0 | -49 | |
| 1 | | | | 1 |
1 | | 50 | 14 | -48 | |
| 3 | | | | 3 |
2 | | 53 | 28 | -45 | |
| 5 | | | | 5 |
3 | | 58 | 42 | -40 | |
| 7 | | | | 7 |
4 | | 65 | 56 | -33 | |
| 9 | | | | 9 |
5 | | 74 | 70 | -24 | |
| 11 | | | | 11 |
6 | | 85 | 84 | -13 | |
| 13 | | | | 13 |
7 | | 98 | 98 | 0 | |
|
|
Table VIII
| δ1 | x |
y | z |
δ2 |
k | |
k2 + 64 |
16k |
k2 - 64 | |
0 | | 64 | 0 | -64 | |
| 1 | | | | 1 |
1 | | 65 | 16 | -63 | |
| 3 | | | | 3 |
2 | | 68 | 32 | -60 | |
| 5 | | | | 5 |
3 | | 73 | 48 | -55 | |
| 7 | | | | 7 |
4 | | 80 | 64 | -48 | |
| 9 | | | | 9 |
5 | | 89 | 80 | -39 | |
| 11 | | | | 11 |
6 | | 100 | 96 | -28 | |
| 13 | | | | 13 |
7 | | 113 | 112 | -15 | |
|
Only eight triples, (x,y,z), were used in the calculation for the Diophantine equation x2 − y2 = 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 II. Go to Part I. Go to Part III.
Go back to homepage.
Copyright © 2020 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com