The Diophantine Equation x² − 5y² = z² (Part VIII)

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		6	2	4
	15				15
2		21	4	19
	25				25
3		46	6	44
	35				35
4		81	8	79
	45				45
5		126	10	124
	55				55
6		181	12	179
	65				65
7		246	14	244

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

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		14	6	-4
	15				15
2		29	12	11
	25				25
3		54	18	36
	35				35
4		89	24	71
	45				45
5		134	30	116
	55				55
6		189	36	171
	65				65
7		254	42	236

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

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		30	10	20
	15				15
2		45	20	-5
	25				25
3		70	30	20
	35				35
4		105	40	55
	45				45
5		150	50	100
	55				55
6		205	60	155
	65				65
7		270	70	220

Table VI
	δ₁	x	y	z	δ₂
k		5k² + 36	12k	5k² - 36
0		36	0	-36
	5				5
1		41	12	-31
	15				15
2		66	24	-16
	25				25
3		81	36	9
	35				35
4		116	48	44
	45				45
5		161	60	89
	55				55
6		216	72	144
	65				65
7		281	84	209

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		54	14	-44
	15				15
2		69	28	-29
	25				25
3		94	42	-4
	35				35
4		129	56	31
	45				45
5		174	70	76
	55				55
6		229	84	131
	65				65
7		294	98	196

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

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 VIII. Go back to Part VII.

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The Diophantine Equation x2 − 5y2 = z2 (Part VIII)

A Method of Generating Triples from Novel Equations

The Diophantine Equation x² − 5y² = z² (Part VIII)