The Diophantine Equation x² + 3y² = z² (Part V)

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		2	2	4
	9				9
2		11	4	13
	15				15
3		26	6	28
	21				21
4		47	8	49
	27				27
5		74	10	76
	33				33
6		107	12	109
	39				39
7		146	14	148

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

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		-6	6	12
	9				9
2		3	12	21
	15				15
3		18	18	36
	21				21
4		39	24	57
	27				27
5		66	30	84
	33				33
6		99	36	117
	39				39
7		138	42	156

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

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		-22	10	28
	9				9
2		-13	20	37
	15				15
3		2	30	52
	21				21
4		23	40	73
	27				27
5		50	50	100
	33				33
6		83	60	133
	39				39
7		122	70	172

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

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		-46	14	52
	9				9
2		-37	28	61
	15				15
3		-22	42	76
	21				21
4		-1	56	97
	27				27
5		26	70	124
	33				33
6		59	84	157
	39				39
7		98	98	196

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

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 V. Go to Part IV. Go to Part VI.

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The Diophantine Equation x2 + 3y2 = z2 (Part V)

A Method of Generating Triples from Novel Equations

The Diophantine Equation x² + 3y² = z² (Part V)