The Diophantine Equation x² − Dy² = z² (Part II)

A Method of Generating Triples from Novel Equations

A new, general method for generating triples of the form (x,y,z) for the Diophantine equation x² − Dy² = z², where D, is any integer greater than zero is being introduced here. The method produces a set of novel second order equations which generates Diophantine triples. These equations are listed below using the following table heading format:

where a is a generating number starting at zero, D is any integer greater than zero and (n², 2n) are values taken from table A:

Since there are an infinite number of (n², 2n) integers there are also an infinite number of tables which can be tabulated with their accompanying equations. A second sequential access is also possible using the δ₁ and δ₂ columns whose δ₁δ₁ and δ₂δ₂ are both 2 for all the tables. In addition,the triples which are generated may be present in primitive or non primitive form, and these non primitives can be converted to primitive forms by dividing out any common factors. When D equals one the Diophantine equation takes the form of x² − y² = z². The calculations for this equation are shown in Tables I thru VIII. Note tha z may be negative at times, the reason being that the square root of z² can take on the values ±z as is verified by the δ₂ differences.

Next comes Tables III and IV.

Next comes Tables V and VI.

Finally we have Tables VII and VIII.

Only eight triples, (x,y,z), were used in the calculation for the Diophantine equation x² − y² = 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 or using δ₁ and δ₂ to access the (x,y,z)s sequentially.

This concludes Part II. Go to Part I. Go to Part III.

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