The Diophantine Equation x² + 4y² = z² (Part IX)

A Method of Generating Triples from Novel Equations

This page is a method for generating triples of the form (x,y,z) for the Diophantine equation x² + 4y² = z².

This new method is a way of producing triples from a set of novel equations that may be used to access the triples, x, y and z. The x, y and z as well as the equations are listed in the table headings below according to the following format:

where a is a generating number starting at zero, D is any integer greater than zero and the (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 8 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.

Since x² + 4y² = z² and x² − 4y² = z² give identical entries in the tables except that the z columns are swapped with the x columns only the former equation will be used with the numbers generated in the tables. In addition, substituting the values in row δ₁ = 3 of table I into the x² + 4y² = z² equation, it can be seen that 35² + 4(6²) = 37².

Next comes Tables III and IV.

Finally Tables V and VI.

Only six triples, (x,y,z), were used in the calculation for the Diophantine equation x² + 5y² = 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 IX. Go to Part IX.
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