Sequences of Pythagorean Multiples (Part XVb)

Use of the Diophantine Equation x² ± D(∑ y_j²) = z² for Generating the Sequences

In Part XII Diophantine triples using the Diophantine equation x² + D(∑ y_j²) = z² was used to produce four examples. This section will show how to create consecutive triples of the Pythagorean equation to generate sequences where the a, b and c of the Pythagorean equation a² + b² = c² are integral. In addition, new modified arithmetic progression and random access methods will be used to calculate the triplets.

We first produce all triples where all the a_j are equal and where the subscript j increases consecutively from 1 to k. This is shown in Table I where for example all the a_js are equal to 5. Initially the n² is greater than the (a_j)² value and the method consists of substituting the values from the appropriate columns of Table I into the Pythagorean equation to obtain the value of z²:

a² + b² = c²
(D∑(a_j²) − n²)² + D∑(y_j)² = z²

Table I (D = 1, n= 6)
					x	y₁	y₂	y₃	...	y_k	z
a₁	a₂	a₃	...	a_k	D(a₁² + a₂² + a₃² + ... + a_k²) − n²	2na₁	2na₂	2na₃			D(a₁² + a₂² + a₃² + ... + a_k²) + n²
5	0	0			(25) − 36 = −11	60	0	0			61
5	5	0			(25 + 25) − 36 = 14	60	60	0			86
5	5	5			(25 + 25 + 25) − 36 = 39	60	60	60			111
⋮
5	5	5	...	5	(25 + 25 + 25 + ...) − 36 = 25k −36	60	60	60	...	60	25k + 36

The calculations were done for k = 1 to 16 according to Table I but all subsequent x values for Table II were obtained by multiplying the (a_j)² (in this case 25) with multiples of m² = 1,4,9,16,... then subtracting 36 as exhibited in Table I. In addition, the initial y value from Table I was used in the arithmetic progression formula to obtain further values of y:

a_k = a_i + d(k − 1)) (A1)

On inspection of Table II we see that the common difference between adjacent x and z values is δ₁ and that the difference between adjacent δ₁s is δδ = 50. Moreover, when we get to the end of Table II we can continue generating new x and z² values by using the various delta (δ) differences.

Table II (δy=60)
δδ	δ₁	x	y	z	δ₁	δδ
		−11	60	61
	75				75
50		64	120	136		50
	125				125
50		189	180	261		50
	175				175
50		364	240	436		50
	225				225
50		589	300	661		50
	275				275
50		864	360	936		50
	325				325
50		1189	420	1261		50
	375				375
50		1564	480	1636		50
	425				425

Note that the even rows give non-primitive triangles and must, consequently, be divided by two. Also note that the first entry for x is negative and this value will throw off the first δ₁ and δδ differences for this particular x. The x and z values may alternatively be obtained using equation (A2) as long as z is not of the form non-integral √z, since this must require a different way of generating the common difference.

b_k+1 = b_k + (a_i + (k − 1)d) (A2)

where a_i is the initial δ₁ value, d is the common difference δδ and b_k are the variables x or z. The b_k+1, thus, obtained in the first step becomes the b_k of the next step. On the other hand, the values for y are obtained using the regular arithmetic progression formula (A1) as was mentioned above.

We place the initial numbers for b_ks of −11 and 61 along with the values for a_i = 75 and d = 50 into equation (A2) and generate the two Tables IIIx and IIIz. We see that except for the first b_k number, the values are in agreement with those of Table II.

Table IIIx (d = δδ = 50)
b_k+1 = b_k + (a_i + (k − 1)d)
b₂ = −11 + (75 + 0(50)) = 64
b₃ = 64 + (75 + 1(50)) = 189
b₄ = 189 + (75 + 2(50)) = 364
b₅ = 364 + (75 + 3(50)) = 589
b₆ = 589 + (75 + 4(50)) = 864
b₇ = 864 + (75 + 5(50)) = 1189
b₈ = 1189 + (75 + 6(50)) = 1564
b₉ = 1564 + (75 + 7(50)) = 1989

Table IIIz (d = δδ = 50)
b_k+1 = b_k + (a_i + (k − 1)d)
b₂ = 61 + (75 + 0(50)) = 136
b₃ = 136 + (75 + 1(50)) = 261
b₄ = 261 + (75 + 2(50)) = 436
b₅ = 436 + (75 + 3(50)) = 661
b₆ = 661 + (75 + 4(50)) = 936
b₇ = 936 + (75 + 5(50)) = 1261
b₈ = 1261 + (75 + 6(50)) = 1636
b₉ = 1636 + (75 + 7(50)) = 2061

Generating Pythagorean Multiples Via Random Access

Though consecutive Pythagorean triples can be obtained via this manner, a new way of generating the x and z values has been produced (Part XIX). Since each triplet is k² distance from each other, the equations are modified to k²x_i + Δ ∕2(k²−1) for the value of x and k²x_i + Δ ∕2(k²+1) for the value of z. Δ corresponds to the difference z − x in Table III and x_i corresponds to the initial value of x in the table and k can be any value greater than zero. Thus, for example:

Δ = 72, k = 9 and x_i = −11
x: −11 × 81 + 36 × 80 = 1989
z: −11 × 81 + 36 × 82 = 2061

Identical to the last b₉ values of Tables IIIx and IIIz. If we take k = 31 we get:

Δ = 72, k = 31 and x_i = −11
x: −11 × 961 + 36 × 960 = 23989
z: −11 × 961 + 36 × 962 = 24061
z² − x² = 24061² −23989² = 1860²

Thus, we can access randomly Pythagorean triples of this type via the use of these equations.

This concludes Part XVb. Go to Part XVc for using pair of a_js. Go to Part XVI where the zs are not perfect squares.

Go back Part XV or to Part XIII. Go back to homepage.

Sequences of Pythagorean Multiples (Part XVb)

Use of the Diophantine Equation x2 ± D(∑ yj2) = z2 for Generating the Sequences

Generating Pythagorean Multiples Via Random Access

Use of the Diophantine Equation x² ± D(∑ y_j²) = z² for Generating the Sequences