The Diophantine Equation 2y² − x² = z² (Part I/IIA)

A Method for Generating Random Access Equations

Generation of Tables of Equations

Math and computer methods in Part I and Part II were used to solve the Diophantine equation 2y² − x² = z² for various triple values of (a₁,b₁,c₁) but performed in a sequential manner. However, these methods require longer and longer calculation times if the desired triple, (a,b,c), is composed of extremely large numbers.

The method in this section uses the triples (a₁,b₁,c₁) from column 2, 3 and 4 of Part I and Part II slightly modified to produce Tables I and II here, while Tables III thru VIII of triples were generated using the same computer program as in Parts I and II but not published online. The method is identical to those of the just published online papers Part III and Part IV where the purpose was to generate equations from the data to provide a method of obtaining new triples via random access. The method, repeated here, starts by setting up the table headings with three equations, constructed from the tabulated data, and placed beneath their appropriate a, b and c variables.

The table headings contain the equations generated from the data for each of the (a₁,b₁,-c₁) and (a₁,-b₁, c₁) triples, shown in the captions for all the tables below, and are generally of the form:

where k is a counter starting at zero, D is the coefficient of y and (n, m) are values taken from the table below so that whenever n = j² then m = 2j for all j > 0:

Since there are an infinite number of (n, m) integers there are also an infinite number of tables which can be tabulated with their accompanying equations; equations whose purpose is to generate the triples either by random or sequential access. In addition, a substantial number of triples are composed of numbers which can be simplified further via prime division.

The utility of these equations is that they are useful for generating the triples via a random access of k unlike Parts I and II which are sequential methods for generating triples. However, in order to generate these new equations we need to use the sequential method to generate a small number of triples which are then used to generate the requisite new equations, the reason for the tables below. The caption for each table pertains to the initial triple value (a₁,b₁,c₁) used in the computer program calculation. Note that the δ₁δ₁ and δ₂δ₂ are 14 and 2, respectively, for all the tables. In addition, a substantial number of triples are composed of numbers which can be simplified further via prime division.

Next comes Tables III and IV.

Next comes Tables V and VI.

Finally we have Tables VII and VIII.

Only eight triples, (a₁,b₁,c₁), were used in the calculation for the Diophantine equation 2y² − x² = z² to get an idea of its versatility. Note that the tables produced can be expanded downward indefinitely. In addition, using the tables one can generate the appropriate equations that are useful for generating any triple in a table using a random access method which obviates the need to go through a sequential time consuming process.

This concludes Part I/II. Go back to Part I or Part II.

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