Sequences of Pythagorean Multiples (Part XVc)

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 of the Pythagorean equation a² + b² = c² are integral. For example, we will produce triples from pairs of a_j where the first a_j is a prime and the second a_j+1 is a multiple of that prime, and where the subscript j is added pairwise, consecutively, from (1,2) to (k,k+1). This is shown in Table I where, for example, the pair of a_js are equal to 3 and 6. In addition, both D and n have a value of one. Since the equation for z is quite lengthy it has been calculated outside the table to a value of z₁ and that value entered into the table.

Thus, the method consists of substituting the values from the appropriate columns of Table I into the Pythagorean equation to obtain the value of z²:

To obtain a y² whose value is a perfect square and whose square root is an integer, calculations are carried out from k = 1 up to a particular value of k which produces this result. We can take this integer and use it to generate the x and z values as well as the next y integer in the sequence. Using values of k we find that the first integral triplet found is when k = 5, where the y² value is 900 and whose square root is 30. The next value of y² is 3600 with a square root of 60. We can calculate every y value after that using the regular arithmetic progression formula where both a_i and d are 30.

On inspection of Table II we see that the common difference between adjacent x and z values is δ₁ and that the difference between adjacent δ₁s is δδ = 450. 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. In addition, if we know y² we can divide this number by four to get the x + 1 value (column 1), and, consequently, the x value (column 3). Furthermore, if we were not given y² but were given the value in column 2 we could easily calculate its value.

Note that the odd rows give non-primitive triangles and must, consequently, be divided by two. 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.

and 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) as was described above.

We place the initial numbers for b_ks of 224 and 226 along with the values for a_i = 675 and d = 450 into equation (A2) and generate the two Tables IIIx and IIIz. We see that these b_k values are in agreement with those of Table II.

This concludes Part XVc. Go to Part XVI where the zs are not perfect squares.

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