The Diophantine Equation x² ± D(∑ y_j²) = z² (Intro)

A Method of Generating Multiples from Novel Equations

In mathematics a Diophantine equation is an equation, typically a polynomial equatiion in two or more unknowns with integer coefficients for which integer solutions are of interest. Diophantine equations may be linear where the exponent = 1 or exponential where the exponents of the unknowns is greater than 1. The number of unknowns in a Diophantine equation is greater than the number of simultaneous equations and the problem is finding integers that solve simultaneously all equations.

The Diophantine equations that we are interested in generating are of the type:

Thus we can generate any multiple of this type of Diophantine equation by running thru all the values of y_j. For instance when k = 1 or k = 2 we obtain, respectively, the following Diophantine triple and quadruple equations:

This new method produces Diophantine multiples from a set of novel equations that are used to access the variables, x, multiple y_j and z. The x, y_j and z as well as the equations are listed in the table headings below according to the following format:

where the a_ks are generating numbers, and D in the general Diophantine equation, x² ± D(∑ y_j²) = z², is the coefficient of the sum of ys and is any number greater than zero. The δ₁ and δ₂ are delta values can be used to access the numbers from one row to the next. Furthermore, the (n², 2n) are values taken from table A:

Furthermore, tables of Diophantine multiples are generated by choosing appropriate generating numbers and appropriate y_k values along with Table A. For instance to generate numbers for Diophantine triples for equation (1), Table B headings are used:

and for Diophantine quadruples of equation (2), Table C headings are used:

Tables of Diophantine triples and quadruples are depicted in Part I-triples, Part I-quadruples, while tables of quintuples and sextuples are depicted in Part X-multiples.

This concludes the Intro.

Go to Part I. Go back to homepage.