The Diophantine Equation x² + 5y² = z² (Part VII)

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 + 5y² = z².

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		5k² - n	mk	5k² + n

where k is a counter starting at zero, the numeral 5 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 10 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		5k² - 1	2k	5k² + 1
0		-1	0	1
	5				5
1		4	2	6
	15				15
2		19	4	21
	25				25
3		44	6	46
	35				35
4		79	8	81
	45				45
5		124	10	126
	55				55
6		179	12	181
	65				65
7		244	14	246

Table II
	δ₁	x	y	z	δ₂
k		5k² - 4	4k	5k² + 4
0		-4	0	4
	5				5
1		1	4	9
	15				15
2		16	8	24
	25				25
3		41	12	49
	35				35
4		76	16	84
	45				45
5		121	20	129
	55				55
6		176	24	184
	65				65
7		241	28	249

Next comes Tables III and IV.

Table III
	δ₁	x	y	z	δ₂
k		5k² - 9	6k	5k² + 9
0		-9	0	9
	5				5
1		-4	6	14
	15				15
2		11	12	29
	25				25
3		36	18	54
	35				35
4		71	24	89
	45				45
5		116	30	134
	55				55
6		171	36	189
	65				65
7		236	42	254

Table IV
	δ₁	x	y	z	δ₂
k		5k² - 16	8k	5k² + 16
0		-16	0	16
	5				5
1		-11	8	21
	15				15
2		4	16	36
	25				25
3		29	24	61
	35				35
4		64	32	96
	45				45
5		109	40	141
	55				55
6		164	48	196
	65				65
7		229	56	261

Next comes Tables V and VI.

Table V
	δ₁	x	y	z	δ₂
k		5k² - 25	10k	5k² + 25
0		-25	0	25
	5				5
1		-20	10	30
	15				15
2		-5	20	45
	25				25
3		20	30	70
	35				35
4		55	40	105
	45				45
5		100	50	150
	55				55
6		155	60	205
	65				65
7		220	70	270

Table VI
	δ₁	x	y	z	δ₂
k		5k² - 36	12k	5k² + 36
0		-36	0	36
	5				5
1		-31	12	41
	15				15
2		-16	24	56
	25				25
3		9	36	81
	35				35
4		44	48	116
	45				45
5		89	60	161
	55				55
6		144	72	216
	65				65
7		209	84	281

Finally we have Tables VII and VIII.

Table VII
	δ₁	x	y	z	δ₂
k		5k² - 49	14k	5k² + 49
0		-49	0	49
	5				5
1		-44	14	54
	15				15
2		-29	28	69
	25				25
3		-4	42	94
	35				35
4		31	56	129
	45				45
5		76	70	174
	55				55
6		131	84	229
	65				65
7		196	98	294

Table VIII
	δ₁	x	y	z	δ₂
k		5k² - 64	16k	5k² + 64
0		-64	0	64
	5				5
1		-59	16	69
	15				15
2		-44	32	84
	25				25
3		-19	48	109
	35				35
4		16	64	144
	45				45
5		61	80	189
	55				55
6		116	96	244
	65				65
7		181	112	309

Only eight triples, (x,y,z), were used in the calculation for the Diophantine equation x² + 5y² = 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 VIII.

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The Diophantine Equation x2 + 5y2 = z2 (Part VII)

A Method of Generating Triples from Novel Equations

The Diophantine Equation x² + 5y² = z² (Part VII)