The Diophantine Equation x2 − Dy2 = z2

Previous sections, e.g., Part III and Part III produced Diophantine triples and quadruples using the Diophantine equation x² + D(∑ y_j²) = z². The previous section Part X discussed the higher diophantine multiples, viz., 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 according to the following format:

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

The equations are similar in form to the equations for obtaining the 3 legs of the Pythagorean integers X = m² − n², Y = 2mn, and Z = m² − n² except that the generating numbers are not limited to only two numbers. In addition, a₁, a₂ and a₃ 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.

Previous tables have been constructed using values of 1 to 9 as generating numbers. A new table is shown with numbers in these left hand columns greater than zero showing that these columns can accommodate generating numbers greater than zero.

Table I depicts 2 quintuples with their respective a_j, D and n values and where D and n interchange their values.

Table I (D = 3, n= 4; D = 4, n = 3)
			x	y₁	y₂	y₃	z
a₁	a₂	a₃	D(a₁² + a₂² + a₃²) - n²	2na₁	2na₂	2na₃	D(a₁² + a₂² + a₃²) + n²
17	19	23	3(289 + 361 + 529) − 16 = 3521	136	152	184	3(289 + 361 + 529) + 16 = 3553
17	19	23	4(289 + 361 + 529) − 9 = 4707	102	124	158	4(289 + 361 + 529) + 9 = 4725

x² + D(y₁² + y₂² + y₃²) = z²
3521² + 3(136² + 152² + 184²) = 12397441 + 3(18496 +23104 + 33856) = 12623809 = 3553²
4707² + 4(102² + 114² + 138²) = 22155849 + 4(10404 +12996+ 19044) = 263225625 = 4725²

The Equivalence of the Left and Right Hand Sides of Equation x² + D(∑ y_j²) = z²

We can show that the left and right hand sides of the general equation x² + D(∑ y_j²) = z² are equal by looking first at an equation that contains only one a_j:

(Da₁² − n²)² + D(2na₁)² = (Da₁² + n²)²
D²a₁⁴ − 2Dn²a₁² + n⁴ + 4Dn²a₁² = (Da₁² + n²)²
D²a₁⁴ + 2Dn²a₁² + n⁴ = (Da₁² + n²)²

which shows that the left hand side equals the right hand side. If we add more a_j for example an a₂ we get:

(D(a₁² + a₂²) − n²)² + D[(2na₁)² + (2na₂)²] = [D(a₁² + a₂²) + n²]²
D²(a₁² + a₂²)² − 2D(n²a₁² + n²a₂²) + n⁴ + 4D(n²a₁² + n²a₂²) = [D(a₁² + a₂²) + n²]²
D²(a₁² + a₂²)² + 2D(n²a₁² + n²a₂²) + n⁴ = [D(a₁² + a₂²) + n²]²

Therefore, the addition of more a_j will result in an equation containing more and more a_j with the proviso that the left hand side of the equation will always equal the right hand side.

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

A Method of Generating Multiples from Novel Equations

The Equivalence of the Left and Right Hand Sides of Equation x² + D(∑ y_j²) = z²

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

A Method of Generating Multiples from Novel Equations

The Equivalence of the Left and Right Hand Sides of Equation x2 + D(∑ yj2) = z2

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

The Equivalence of the Left and Right Hand Sides of Equation x² + D(∑ y_j²) = z²