Sequences of Pythagorean Multiples (Part XV)

Table of Pythagorean triples from Diophantine Equation x² ± D(∑ y_j²) = z²

In this section fully integral Pythagorean multiples derived from a_js of 5 and 7 generated by the method of Part XIV will be displayed. The section will show how to create consecutive triples of the Pythagorean equation to generate sequences where the a, b and c are integral numbers. In addition, a new modified arithmetic progression method as well as a new random access method to generate the triples will be shown. Only the sequence of Pythagorean triples will be shown without going into the actual calcuations performed in Part XIV. Table I refers to those triples obtained from a_j of 5 and Table II to those of a_j of 7.

Table I (δy=10)
δδ	δ₁	x	y	z	δ₁	δδ
		24	10	26
	75				75
50		99	20	101		50
	125				125
50		224	30	226		50
	175				175
50		399	40	401		50
	225				225
50		624	50	626		50
	275				275
50		899	60	901		50
	325				325
50		1224	70	1266		50
	375				375
50		1599	80	1601		50
	425				425

Table II (δy=14)
δδ	δ₁	x	y	z	δ₁	δδ
		48	14	50
	147				147
98		195	28	197		98
	245				245
98		440	42	442		98
	343				343
98		783	56	785		98
	441				441
98		1224	700	1226		98
	539				539
98		1763	84	1765		98
	637				637
98		2400	98	2402		98
	735				735
98		3135	112	3137		98
	833				833

Normally the common differences are found via the equation:

a_k = a_i + (k − 1)d (A1)

Since there are 2 common differences, one which is variable and the other constant, the x and z values may alternatively be obtained using the modified equation (A2) as long as z is not of the form √z, since this would require a different method.

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).

For an a_j = 5 we place the initial numbers for b_ks of 99 and 101, obtained in equation (A2), and perform the calculations for the various b_ks in Tables IIIx and IIIz. On inspection we see that these b_k values are in agreement with those of Table I.

Table III x (d = δδ = 50)
b_k+1 = b_k + (a_i + (k − 1)d)
b₂ = 24 + (75 + 0(50)) = 99
b₃ = 99 + (75 + 1(50)) = 224
b₄ = 224 + (75 + 2(50)) = 399
b₅ = 399 + (75 + 3(50)) = 624
b₆ = 624 + (75 + 4(50)) = 899
b₇ = 899 + (75 + 5(50)) = 1224
b₈ = 1224 + (75 + 6(50)) = 1599
b₉ = 1599 + (75 + 7(50)) = 2024

Table III z (d = δδ = 50)
b_k+1 = b_k + (a_i + (k − 1)d)
b₂ = 26 + (75 + 0(50)) = 101
b₃ = 101 + (75 + 1(50)) = 226
b₄ = 226 + (75 + 2(50)) = 401
b₅ = 401 + (75 + 3(50)) = 226
b₆ = 226 + (75 + 4(50)) = 901
b₇ = 901 + (75 + 5(50)) = 1266
b₈ = 1266 + (75 + 6(50)) = 1601
b₉ = 1601 + (75 + 7(50)) = 2026

For an a_j = 7 we place the initial numbers for b_ks of 48 and 50, obtained in equation (A2), and perform the calculations for the various b_ks in Tables IVx and IVz. And we see that these b_k values are in agreement with those of Table II.

Table IVx (d = δδ = 98)
b_k+1 = b_k + (a_i + (k − 1)d)
b₂ = 48 + (147 + 0(98)) = 195
b₃ = 195 + (147 + 1(98)) = 440
b₄ = 440 + (147 + 2(98)) = 783
b₅ = 783 + (147 + 3(98)) = 1224
b₆ = 1224 + (147 + 4(98)) = 1763
b₇ = 1763 + (147 + 5(98)) = 2400
b₈ = 2400 + (147 + 6(98)) = 3135
b₉ = 3135 + (147 + 7(98)) = 3968

Table IVz (d = δδ = 98)
b_k+1 = b_k + (a_i + (k − 1)d)
b₂ = 50 + (147 + 0(98)) = 197
b₃ = 197 + (147 + 1(98)) = 442
b₄ = 442 + (147 + 2(98)) = 785
b₅ = 785 + (147 + 3(98)) = 1226
b₆ = 1226 + (147 + 4(98)) = 1765
b₇ = 1765 + (147 + 5(98)) = 2402
b₈ = 2402 + (147 + 6(98)) = 3137
b₉ = 3137 + (147 + 7(98)) = 3970

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, for Tables III:

Δ = 2, k = 9 and x_i = 24
x: 81 × 24 + 80 = 2024
z: 81 × 24 + 82 = 2026

Identical to the last b₉ values of Tables IIIx and IIIz. If we take k = 25 we get:

Δ = 2, k = 25 and x_i = 24
x: 625 × 24 + 624 = 15624
z: 625 × 24 + 626 = 15626
z² − x² = 15626² − 15624² =250²

And, for example, for Tables IV:

Δ = 2, k = 9 and x_i = 24
x: 81 × 48 + 80 = 3968
z: 81 × 48 + 82 = 3970

Identical to the last b₉ values of Tables IVx and IVz. If we take k = 25 we get:

Δ = 2, k = 25 and x_i = 24
x: 625 × 48 + 624 = 30624
z: 625 × 48 + 626 = 30626
z² − x² = 30626² − 30624² = 350²

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

This concludes Part XV. The next section (Part XVI) will show a novel way of obtaining sequences of partially integral Pythagorean triples where the square roots of the z values are non-perfect squares and must require a different way of generating the common difference.

Go to Part XVb for an a_j of 5 with initially n² is greater than 25.
Go back to Part XIV. Go back to homepage.

Sequences of Pythagorean Multiples (Part XV)

Table of Pythagorean triples from Diophantine Equation x2 ± D(∑ yj2) = z2

Generating Pythagorean Multiples Via Random Access

Table of Pythagorean triples from Diophantine Equation x² ± D(∑ y_j²) = z²