Sequences of Pythagorean Multiples (Part XIX)

New Method for Random Accessing of x, y² and z² Variables

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 y and z values are non-perfect squares. in addition, three new equations have been produced that can be used to access larger Pythagorean triplets and do so via random access instead of having to trudge through one set of triples after another.

We first produce all triples where the a_j, e.g., (a₁,a₂,a₃) can either take on the values (3,5,7) or (9,1,1) where these subscript j increases consecutively from 1 to k. Both sets of values afford the same ∑ of a_j². This is shown in Table I where, for example, all the three a_js are equal to 3,5,7. 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 9, the following results are obtained and listed in Tables IIa and IIb, where the δy² and the δz² values are used to avoid non-integral values of y and z. In addition, the δ_x between adjacent x values is 83 which may also be obtained from the arithmetic progression formula using 82 as the initial number, a_i, and 83 as the common difference, d:

We split the results into two tables in order to obtain the differences of y without overcrowding Table IIa. In addition, we have Table IIc where a) initially z = √7056 = 84 and where b) each row is calculated using the initial value x_i = 82 in the random access equations of Part XIV. While I may be jumping the gun, these equations were derived from equation (B1) below.

As shown in Table IIc the values of y² calculated using the random access equations are identical to those from different rows of Table IIa where the zs are non integer values. The difference between Table IIa and Table IIc is that the rows of table IIa are derived using k as the multiplicand while those of Table IIc are derived using k².

The y² and z² values may be obtained using equation (A2) as previously shown in Part XV.

where a_i is the initial δ_y² or δ_z² value, d is the common difference δδy² or δδz²and b_k are the variables y² or z². The b_k+1, thus, obtained in the first step becomes the b_k of the next step.

A New Way of Randomly Accessing the x, y² and z² Values

Previously we have been generating the values of x, y² and z² in a consecutive manner using equation A2 or by the delta (δ) differences in Table IIa. A new method of generating the x, y² or z² values of any Pythagorean triplets of this type via random access was, therefore, warranted and thus three equations were created using a variable k to generate new x, y² or z² values without having to calculate each value consecutively.

and where the x_i, y_i² and z_i² are the initial values from the table and where the d_y² and d_z² are the two δδ values.

Examples of Triplets Using the Three Equations

Three examples using the above equations with different values for k are shown below:

Which confirms the last row values of Table IIa. The values for k = 20 and 36 are as follow:

Note that the three new equations are only good where the differences are of the type δδ. No equations for δδδ or higher differences have to date been found.

This concludes Part XIX. Go to Part XX for random access of a few known Pythagorean triples.
Go back to Part XVIII. Go back to homepage.