The Diophantine Equation x² ± D(∑ y_j²) = z² (Multiples) (Part X)

A Method of Generating Multiples from Novel Equations

Previous sections, e.g., Part III and Part III produced diophantine triples and quadruples using the Diophantine equation x² + D(∑ y_j²) = z². This section continues the discussion of the higher diophantine multiples, for instance, the quintuples, sextuples using the same equation with more y_j values.

This new method produces multiples from a set of novel equations generating the variables x, y₁, y₂, ..., y_m and z, where there occur multiple variables of y. The first example using the variables x, y₁, y₂, y₃ and z for the quintuples as well as the equations listed in table A1 and A2 headings below according to the following formats:

Table A1
			δ₁	x	y₁	y₂	y₃	z	δ₂
a₁	a₂	a₃		D(a₁² + a₂² + a₃²) - n²	2na₁	2na₂	2na₃	D(a₁² + a₂² + a₃²) + n²

Table A2
				δ₁	x	y₁	y₂	y₃	y₄	z	δ₂
a₁	a₂	a₃	a₄		D(a₁² + a₂² + a₃² + a₄²) - n²	2na₁	2na₂	2na₃	2na₄	D(a₁² + a₂² + a₃² + a₄²) + n²

where a₁, a₂, a₃ and a₄ are generating numbers, D is any integer greater than zero and the (n², 2n) are values taken from table B:

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. In addition, the multiples which are generated may be present in primitive or non primitive form, where non primitives may be converted to primitive forms by dividing out any common factors.

Table I depicts 3 quintuples with their respective a_j, D and n values:

Table I (D = 2, n² = 1; D = 3, n² = 4; D = 5, n² = 9)
			x	y₁	y₂	y₃	z
a₁	a₂	a₃	D(a₁² + a₂² + a₃²) - n²	2na₁	2na₂	2na₃	D(a₁² + a₂² + a₃²) + n²
1	1	1	2(1 + 1 + 1) − 1 = 5	2	2	2	2(1 + 1 + 1) + 1 = 7
3	4	5	3(9 + 16 + 25) − 4 = 146	12	16	20	3(9 + 16 + 25) + 4 = 154
8	7	6	5(64+ 49 + 36) − 9 = 736	48	42	36	5(64 + 49 + 36) + 9 = 754

Plugging these values into the equation:

x² + D(y₁² + y₂² + y₃²) = z²
25 + 2(4 + 4 + 4) = 49
21316 + 3(144 + 256 + 400) = 23716 = 154²
541696 + 5(2304 + 1764 + 1296) = 568516 = 754²

confirming the z values of Table I.

Table II depicts 3 sextuples with their respective a_j, D and n values:

Table II (D = 4, n² = 9; D = 3, n² = 4; D = 5, n² = 9)
				x	y₁	y₂	y₃	y₄	z
a₁	a₂	a₃	a₄	D(a₁² + a₂² + a₃² + a₄²) - n²	2na₁	2na₂	2na₃	2na₄	D(a₁² + a₂² + a₃² + a₄²) + n²
2	2	2	2	4(4 + 4 + 4 + 4) − 9 = 55	12	12	12	12	4(4 + 4 + 4 + 4) + 9 = 73
3	4	5	6	3(9 + 16 + 25 + 36) − 4 = 254	12	16	20	24	3(9 + 16 + 25 + 36) + 4 = 262
8	7	6	5	5(64 + 49 + 35 + 25) − 9 = 861	48	42	36	30	5(64 + 49 + 35 + 25) + 9 = 879

Plugging these values into the equation:

x² + D(y₁² + y₂² + y₃² + y₄²) = z²
3025 + 4(144 + 144 + 144 + 144) = 5329 = 73²
64516 + 3(144 + 256 + 400 + 576) = 68644 = 262²
741321 + 5(2304 + 1764 + 1296 + 900) = 772641 = 879²

confirming the z values of Table II. One need not stop at the multiples in this section but can continue on to the septuples, octatuples, and so forth, all of which can be generated via this method.

This concludes Part X. Go back to Part IX.

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The Diophantine Equation x2 ± D(∑ yj2) = z2 (Multiples) (Part X)

A Method of Generating Multiples from Novel Equations

The Diophantine Equation x² ± D(∑ y_j²) = z² (Multiples) (Part X)