Sequences of Pythagorean Multiples (Part XVI)

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

In Part XIII an alternative method was used to generate a couple of partial integral Pythagorean triples, while Part XIV was used to generate all integral Pythagorean triples. In this section the method of Part XIV will be used to generate a sequence of partial integral triples where the z value is a non-perfect square and using the method of Part XIII where n² is added instead of being subtracted.

We first produce all triples where all the a_j are equal and where these generating numbers a_j increase consecutively from 1 to k. This is shown in Table I where for example all the a_js are equal to 3. Since the entire equation for z is too big to fit in the cell, it has been set equal to Root and this variable placed within the cell. The last root is some variable, N_k, whose square root is again a non-perfect square.

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 10 as the initial number and 9 as the common difference:

On inspection the x column shows that these numbers are the identical numbers from the z column of Tables IIa and IIB of Part XIV, while inspection of the y 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 adding 1 as exhibited in Table I.

On inspection of Table III we see that the common difference between adjacent x values is δ₁ and that the difference between adjacent δ₁s is δ₁δ₁ = 18. The common difference between adjacent z² values, however, is not 18 but is obtained thru four levels of δ differences. In addition, we use the z² to avoid working with square roots. 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 III contains additional triples obtained using δ1 and δ₁δ₁ columns. In addition, the first row (1,0,1) was added by continuing backwards from the third row.

The x values may be obtained using equation (A2):

where a_i is the initial δ₁ value, d is the common difference δδ and b_k is the variable x. 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).

The z values, on the other hand, must be obtained directly from the regular z² = x² + y². By using the delta (δ) differences in the last four columns, the fourth level common difference δδδδ was found to be 1944.

Alternatively, to calculate the values of x using equation (A2), we place the initial numbers for b_k of 10 in equation (A2) and perform the calculations for the various b_ks in Tables IIIx. The b_k values are in perfect agreement with those of Table III.

Alternatively we can calculate the δδδδ values using the light green or key lime green cell values of Table V.

We can take any z²_{row r1}, subtractthe cell below, z²_{row r+1}, and then diagonally, subtract then add, subtracting then adding until the δδδδ is reached.

To calculate a z² starting from a δδδδ we add values diagonally down until the z² column is reached then dropping down one cell gives a new value. For example, adding up the diagonal light blue elements of the table we eventually reach a new z² after dropping down one cell:

This concludes Part XVI. Go to Part XVII.
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