Sequences for Pythagorean Multiples

In this section we will show how to create consecutive triples of the Pythagorean equation a² + b² = c² using the random access equations employed in Part XIV. Five Pythagorean triples listed in Recreations in the Theory of Numbers by Albert H. Beiler (Pages 106 and 123) (1964) will be used to generate new multiples which can be employed in the generation of Pythagorean sequences.

Table I contains the five triples and their differences Δ = z − x and half differences Δ /2 which are the squares of whole numbers.

Generating Pythagorean Multiples Via Random Access

Table I
x	y	z	Δ	Δ /2
13	84	85	72	36
697	696	985	288	144
49	1200	1201	1152	576
4059	4060	5741	1682	841
81	3280	3281	3200	1600

x = k²x_i + Δ ∕2(k²−1)
z = k²x_i + Δ ∕2(k²+1)

y =	____ √z² − x²

A series of new Pythagorean triple multiples, where k = 2 and k = 3, were constructed using the new equations and utilizing x_i for the initial value of x. The first triple from Table I is (13, 84, 85):

Δ = 72, k² = 4 and x_i = 13
x = 4 × 13 + 36 × 3 = 160
z = 4 × 13 + 36 × 5 = 232
y = √28224 = 168

k² = 9
x = 9 × 13 + 36 × 8 = 405
z = 9 × 13 + 36 × 10 = 477
y = √63504 = 252

Thus we obtain the two new triple multiples (160, 168, 232) and (405, 252, 477) with the first triplet being reducible to (20, 21, 29).

Δ = 288, k² = 4 and x_i = 697
x = 4 × 697 + 144 × 3 = 3220
z = 4 × 697 + 144 × 5 = 3508
y = √1937664 = 1392

k² = 9
x = 9 × 13 + 36 × 8 = 7425
z = 9 × 13 + 36 × 10 = 7713
y = √4359744 = 2088

Thus we obtain the two new triple multiples (3220, 1392, 3508) and (7425, 2088, 7713) with the first triplet being reducible to (805, 348, 877)

Δ = 1152, k² = 4 and x_i = 49
x = 4 × 49 + 576 × 3 = 1924
z = 4 × 49 + 576 × 5 = 3076
y = √5760000 = 2400

k² = 9
x = 9 × 49 + 576 × 8 = 5049
z = 9 × 49 + 576 × 10 = 6201
y = √12960000 = 3600

Thus we obtain the two new triple multiples (1924, 2400, 3076) and (5049, 3600, 6201) with the first triplet being reducible to (481, 600, 769)

Δ = 1682, k² = 4 and x_i = 4059
x = 4 × 4059 + 841 × 3 = 18759
z = 4 × 4059 + 841 × 5 = 20441
y = √65934400 = 8120

k² = 9
x = 9 × 4059 + 841 × 8 = 43259
z = 9 × 4059 + 841 × 10 = 44941
y = √148352400 = 12180

Thus we obtain the two new triple multiples (18759, 8120, 20441) and (43259, 12180, 44941).

Δ = 3200, k² = 4 and x_i = 81
x = 4 × 81 + 1600 × 3 = 5124
z = 4 × 81 + 1600 × 5 = 8324
y = √43033600 = 6560

k² = 9
x = 9 × 81 + 1600 × 8 = 13529
z = 9 × 81 + 1600 × 10 = 16729
y = √96825600 = 9840

Thus we obtain the two new triple multiples (18759, 8120, 20441) and (13529, 9840, 16729).

Moreover, we can randomly access and construct an infinite number of Pythagorean triples via the use of these new equations and, thereby, produce sequences of triples for each initial set of Pythagorean triples all based on the value of k before squaring.

This concludes Part XX. Go to Part XXI for generating sequences of primitive and non-primitive Pythagorean triples.
Go back Part XIX. Go back to homepage.

Generation of Pythagorean Triples via Random Access of Known Triples (Part XX)

Generating Pythagorean Multiples Via Random Access