Sequences of Pythagorean Multiples (Part XIV)

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 are integral numbers. In addition,a new modified arithmetic progression method and a new random access method to generate the triples will be shown. 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 3.

When the calculations are done for k = 1 to 16, the following results are obtained and listed in Tables IIa and IIb, where the δy² values are used to avoid non-integral values of y. In addition, the δ₁ between adjacent x and z values is 9 which may also be obtained from the arithmetic progression formula using 8 as the initial number, a_i, and 9 as the common difference, d:

Inspection of the y² column shows, that only the bold/italized values give y integral values of 6, 12, 18 and 24 and where their common difference is 6. We can extract these particular italized triples from these two tables and generate a sequence of integral Pythagorean triples (Table III). Alternatively, the xs can be obtained by multiplying (a_j)² (in this case 9) with multiples of m² = 1,4,9,16,... then subtracting 1 as exhibited in Table I.

On inspection we see that the common difference between adjacent x and z values is δ₁ and that the difference between adjacent δ₁s is δδ = 18. 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.

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.

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).

We place the initial numbers for b_ks of 8 and 10 in equation (A2) and perform the calculations for the various b_ks in Tables IVx and IVz. We see that these b_k values are in agreement with those of Table III.

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:

Identical to the last b₈ values of Tables IIIx and IIIz. If we take k = 25 we get:

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

This concludes Part XIV. The next section (Part XV) will show just the tables of fully integral triples for a_j of 5 and 7 obtained by the above method.

Go back to Part XIII. Go back to homepage.