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
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 (n & m 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 4 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		2a² + 1	2a	2a² - 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
a		2a² + 4	4a	2a² - 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 as 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
a		2a² + 9	6a	2a² - 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
a		2a² + 16	8a	2a² - 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
a		2a² + 25	10a	2a² - 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
a		2a² + 36	12a	2a² - 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
a		2a² + 49	14a	2a² - 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
a		2a² + 64	16a	2a² - 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 V.

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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)