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.

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The Diophantine Equation x2 − Dy2 = z2 (Part II)

A Method of Generating Triples from Novel Equations

The Diophantine Equation x² − Dy² = z² (Part II)