Generating Sequences of Pythagorean Triples (Part XXI)

Primitive and non-Primitive Triples

In this section we will produce tables of consecutive triples of the Pythagorean equation x² + y² = z² by using the regular m, n generating functions we are accustomed to, which are actually simplifications of the equations used in Table I of Part XIV. In addition, a variable Δ is listed in the caption of each of the tables and corresponds to the difference z − x also equal to 2n². This difference is important for generating new Pythagorean multiples from the original triple and may be expanded to generate Pythagorean sequences as was shown in Part XX.

Three triples chosen from Tables III-IV will be used to generate entries into new Pythagorean sequences. The sequences that are produced actually consists of triplet entries further down in the tables since the Δs in each are identical. Therefore, by using these equations we can generate the higher multiples without having to consecutively go through every triple in a table before reaching the desired ones.

What can be gleaned from the tables is that primitive triples are obtained from an (even m, odd n) combination as well as an (odd m, even n) combo. In addition, x can be positive or negative since the difference between z and x will still sum up to positive Δ in either case.

Generating Pythagorean Sequences via Random Access

As described in Part XX multiple triples of primitive or non primitive triplets may be obtained from n² using the following equations with the value of x_i equal to x in the triplet being considered:

One triple from Table III and two from Table IV will be used to generate triplet entries in a Pythagorean sequence. The first set of 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 third triple from Table III is (91, 60, 109):

Thus, we obtain the second and third triple entries, (391, 120, 409) and (891, 180, 909), of the Pythagorean sequence starting with the triple (91, 60, 109). This sequence should, therefore, consist of just primitive entries. For instance, the fourth triplet with k² = 16 is non reducible, viz., (1591, 240, 1609).

The second set of Pythagorean triple multiples, where k = 4 and k = 5, were constructed using the new equations and utilizing x_i for the initial value of x. The second triple from Table IV is (33, 56, 65):

Thus we obtain the fourth and fifth triple entries, (768, 224, 800) and (1209, 280, 1241), in the Pythagorean sequence starting with the triple (33, 56, 65). Note that the fourth entry is reducible to the primitive (24, 7, 25). In addition, the next n² = 36 will also produce a reducible non-primitive triple viz., (1748, 336, 1780) and, therefore, this type of sequence will consist of pairs of adjacent primitive and non-primitive entries.,

The first set of 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 third triple from Table IV is (−7, 24, 25) where the x_i = −7:

Thus, we obtain the second and third triple entries, (20, 48, 52) and (65, 72, 97), of the Pythagorean sequence starting with the triple (−7, 24, 25). Note that the second entry is reducible to the primitive (5, 12, 13). Again, this sequence will consist of pairs of adjacent primitive and non-primitive entries.

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 prior to squaring.

This concludes Part XXI.
Go back Part XX. Go back to homepage.