The Diophantine Equation x² + D(y² + z²) = w² (Part II)

A Method for Generating Quadruples from Novel Equations

This is a continuation of Part I, a method for generating Diophantine quadruples of the form (x,y,z,w) for the Diophantine equation x² + D(y² + z²) = w². The new method produces quadruples from a set of novel equations. The x, y,z and w variables as well as the equations used are listed in the headings of table A using the following format:

Table A
		δ₁	x	y	z	w	δ₂
a	b		D(a² + b²) - n²	2na	2nb	D(a² + b²) + n²

where a and b are generating numbers, D is the coefficient of a² and b² whose value is any number greater than zero and the (n², 2n) are values taken from the following table:

Table B (n² & 2n 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. Access can be either thru the equations or via the use of the δ₁ and δ₂ columns. In addition,the quadruples 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.

The values for (x,y,z,w) are actually variables which can be used in the Diophantine equation x² ± D(y² + z²) = w². Furthermore, since the + and − forms of this equation give identical entries in the tables (z columns are swapped with the w columns), only the + equation will be used with the numbers generated in the tables. In addition, the generating function a has been incremented to 2 for the tables in this section.

To start Table V D is set to 1 and the value of n from Table B above is set to 1.

Table V (D = 1, n = 1)
		δ₁	x	y	z	w	δ₂
a	b		D(a² + b²) - n²	2na	2nb	D(a² + b²) + n²
2	1		4	4	2	6
		3					3
2	2		7	4	4	9
		5					5
2	3		12	4	6	14
		7					7
2	4		19	4	8	21

Table V contains two primitive and two non primitive forms. The non primitive forms (rows 2 and 6) after division by common factors give the Diophantine quadruples (2,2,1,3) and (6,2,3,7) which when plugged into the equation x² + D(y² + z²) = w² gives:

4 + 1(4 + 1) = 9
36 + 1(4 + 9) = 49

All these D = 1 quadruples exhibit the property of being both Pythagorean as well as Diophantine. These quadruples not only are known in the literature but have been generated via similar methods shown here. On the other hand, the Diophantine quadruples with D greater than one may not be. Table VI gives the various values when D = 2.

Table VI (D = 2, n = 1)
		δ₁	x	y	z	w	δ₂
a	b		D(a² + b²) - n²	2na	2nb	D(a² + b²) + n²
2	1		9	4	2	11
		6					6
2	2		15	4	4	17
		10					10
2	3		25	4	6	27
		14					14
2	4		39	4	8	41

Every row is this table consists of primitive quadruples. Squaring and substituting these values into x² + 2(y² + z²) = w² gives:

81 + 2(16 + 4) = 121
225 + 2(16 + 16) = 289
625 + 2(16 + 36) = 729
1521 + 2(16 + 64) = 1681

where the values for the various ws are confirmed. The next two tables VII where D = 3 and VIII where D = 4 are depicted below:

Table VII (D = 3, n = 1)
		δ₁	x	y	z	w	δ₂
a	b		D(a² + b²) - n²	2na	2nb	D(a² + b²) + n²
2	1		14	4	2	16
		9					9
2	2		23	4	4	25
		15					15
2	3		38	4	6	40
		21					21
2	4		59	4	8	61

Squaring and substituting these values into x² + 3(y² + z²) = w² confirms w.

Table VIII (D = 4, n = 1)
		δ₁	x	y	z	w	δ₂
a	b		D(a² + b²) - n²	2na	2nb	D(a² + b²) + n²
2	1		19	4	2	21
		12					12
2	2		31	4	4	33
		20					20
2	3		51	4	6	53
		28					28
2	4		79	4	8	81

Squaring and substituting these values into x² + 4(y² + z²) = w² confirms w.

This concludes Part II. Go to Part III.

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The Diophantine Equation x2 + D(y2 + z2) = w2 (Part II)

A Method for Generating Quadruples from Novel Equations

The Diophantine Equation x² + D(y² + z²) = w² (Part II)