Sequences of Pythagorean Multiples (Part XVIII)

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 and is a continuation of Part XVII.

We first produce all triples where all the a_j are equal and where these 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. 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.

Only those values of y which are multiples of 20 (from Table I) with their appropriate x values are used. The xs are obtained thru the use if Table I or by multiplying (a_j)² (in this case 25) with multiples of m² = 1,4,9,16,... then adding 4 as exhibited in Table I. This results in Table II which the appropriate values of x, y and z² where we use z² avoid non-perfect square roots. On inspection we see that the common difference between adjacent x values is no longer δ₁ but δ₁δ₁ with a value of 50. In addition, the common difference between adjacent z² values (on the right hand side) is not 50 but is obtained thru four levels of δ differences. 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 results for x, y and z² are depicted in Table II along with columns of deltas δ₁δ₁, δδ, δδδ and δδδδ.

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 15000.

Alternatively, to calculate the values of x using equation (A2), we place the initial numbers for b_k of 29 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 II.

Finally there is a trick to calculating the δδδδ values of Table II. For example, we take a z²_{row r1}, subtract the cell below, z²_{row r+1}, and then diagonally, subtract then add, subtracting and adding again and again until the δδδδ is reached (see Table II).

This concludes Part XVIII. Go to Part XIX for a new method for random accessing the various x, y² and z² values.
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