The Diophantine Equation x² − 2y² = z² (Part IV)

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 − 2y² = 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, via random or sequential means. 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
k		2k² + n	mk	2k² - n

where k is a counter starting at zero, the numeral 2 is the coefficient of y and the (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 4 for all the tables. In addition, a substantial number of triples are composed of numbers which can be simplified further via prime division.

Table I
	δ₁	x	y	z	δ₂
k		2k² + 1	2k	2k² - 1
0		1	0	-1
	2				2
1		3	2	1
	6				6
2		9	4	7
	10				10
3		19	6	17
	14				14
4		33	8	31
	18				18
5		51	10	49
	22				22
6		73	12	71
	26				26
7		99	14	97

Table II
	δ₁	x	y	z	δ₂
k		2k² + 4	4k	2k² - 4
0		4	0	-4
	2				2
1		6	4	-2
	6				6
2		12	8	4
	10				10
3		22	12	14
	14				14
4		36	16	28
	18				18
5		54	20	46
	22				22
6		76	24	68
	26				26
7		102	28	98

where Table IIA represents Table II after division by 2 where the ks no longer match up and the δ₁ and δ₂ are no longer relevant.

Table IIA
x	y	z
1	0	-1
3	2	-1
3	2	1
11	6	7
9	4	7
27	10	23
19	6	17
51	14	47

Now for Tables III and IV.

Table III
	δ₁	x	y	z	δ₂
k		2k² + 9	6k	2k² - 9
0		9	0	-9
	2				2
1		11	6	-7
	6				6
2		17	12	-1
	10				10
3		27	18	9
	14				14
4		41	24	23
	18				18
5		59	30	41
	22				22
6		81	36	63
	26				26
7		107	42	89

Table IV
	δ₁	x	y	z	δ₂
k		2k² + 16	8k	2k² - 16
0		16	0	-16
	2				2
1		18	8	-14
	6				6
2		24	16	-8
	10				10
3		34	24	2
	14				14
4		48	32	16
	18				18
5		66	40	34
	22				22
6		88	48	56
	26				26
7		114	56	82

For Tables V and VI.

Table V
	δ₁	x	y	z	δ₂
k		2k² + 25	10k	2k² - 25
0		25	0	-25
	2				2
1		27	10	-23
	6				6
2		33	20	-17
	10				10
3		43	30	-7
	14				14
4		57	40	7
	18				18
5		75	50	25
	22				22
6		97	60	47
	26				26
7		123	70	73

Table VI
	δ₁	x	y	z	δ₂
k		2k² + 36	12k	2k² - 36
0		36	0	-36
	2				2
1		38	12	-34
	6				6
2		44	24	-28
	10				10
3		54	36	-18
	14				14
4		68	48	-4
	18				18
5		86	60	14
	22				22
6		108	72	36
	26				26
7		134	84	60

Finally we have Tables VII and VIII.

Table VII
	δ₁	x	y	z	δ₂
k		2k² + 49	14k	2k² - 49
0		49	0	-49
	2				2
1		51	14	-47
	6				6
2		57	28	-41
	10				10
3		67	42	-31
	14				14
4		81	56	-17
	18				18
5		99	70	1
	22				22
6		121	84	23
	26				26
7		147	98	49

Table VIII
	δ₁	x	y	z	δ₂
k		2k² + 64	16k	2k² - 64
0		64	0	-64
	2				2
1		66	16	-62
	6				6
2		72	32	-56
	10				10
3		82	48	-46
	14				14
4		96	64	-32
	18				18
5		114	80	-14
	22				22
6		136	96	8
	26				26
7		162	112	34

Only eight triples, (x,y,z), were used in the calculation for the Diophantine equation x² − 2y² = 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 IV. Go to Part II. Go to Part IV.

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The Diophantine Equation x2 − 2y2 = z2 (Part IV)

A Method of Generating Triples from Novel Equations

The Diophantine Equation x² − 2y² = z² (Part IV)