The Diophantine Equation x² − 3y² = z² (Part VI)

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, 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		3k² + n	mk	3k² - n

where k is a counter starting at zero, the numeral 3 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 6 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
	3				3
1		4	2	2
	9				9
2		13	4	11
	15				15
3		28	6	26
	21				21
4		49	8	47
	27				27
5		76	10	74
	33				33
6		109	12	107
	39				39
7		148	14	146

Table II
	δ₁	x	y	z	δ₂
k		3k² + 4	4k	3k² - 4
0		4	0	-4
	3				3
1		7	4	-1
	9				9
2		16	8	8
	15				15
3		31	12	23
	21				21
4		52	16	44
	27				27
5		79	20	71
	33				33
6		112	24	104
	39				39
7		151	28	143

Now for Tables III and IV.

Table III
	δ₁	x	y	z	δ₂
k		2k² + 9	6k	2k² - 9
0		9	0	-9
	3				3
1		12	6	-6
	9				9
2		21	12	3
	15				15
3		36	18	18
	21				21
4		57	24	39
	27				27
5		84	30	66
	33				33
6		117	36	99
	39				39
7		156	42	138

Table IV
	δ₁	x	y	z	δ₂
k		2k² + 16	8k	2k² - 16
0		16	0	-16
	3				3
1		-19	8	-13
	9				9
2		28	16	-4
	15				15
3		43	24	11
	21				21
4		64	32	32
	27				27
5		91	40	59
	33				33
6		124	48	92
	39				39
7		163	56	131

For Tables V and VI.

Table V
	δ₁	x	y	z	δ₂
k		2k² + 25	10k	2k² - 25
0		25	0	-25
	3				3
1		28	10	-22
	9				9
2		37	20	-13
	15				15
3		52	30	2
	21				21
4		73	40	23
	27				27
5		100	50	50
	33				33
6		133	60	83
	39				39
7		172	70	122

Table VI
	δ₁	x	y	z	δ₂
k		2k² + 36	12k	2k² - 36
0		36	0	-36
	3				3
1		39	12	-33
	9				9
2		48	24	-24
	15				15
3		63	36	-9
	21				21
4		84	48	12
	27				27
5		111	60	39
	33				33
6		144	72	72
	39				39
7		183	84	111

Finally we have Tables VII and VIII.

Table VII
	δ₁	x	y	z	δ₂
k		2k² + 49	14k	2k² - 49
0		49	0	-49
	3				3
1		52	14	-46
	9				9
2		61	28	-37
	15				15
3		76	42	-22
	21				21
4		97	56	-1
	27				27
5		124	70	26
	33				33
6		157	84	59
	39				39
7		196	98	98

Table VIII
	δ₁	x	y	z	δ₂
k		2k² + 64	16k	2k² - 64
0		64	0	-64
	3				3
1		67	16	-61
	9				9
2		76	32	-52
	15				15
3		91	48	-37
	21				21
4		112	64	-16
	27				27
5		139	80	11
	33				33
6		172	96	44
	39				39
7		211	112	83

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 VI. Go to Part VII.

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The Diophantine Equation x2 − 3y2 = z2 (Part VI)

A Method of Generating Triples from Novel Equations

The Diophantine Equation x² − 3y² = z² (Part VI)