Alternative Method for Calculating Pythagorean Multiples (Part XIII)

Via the Diophantine Equation x² ± D(∑ y_j²) = z²

In the previous section Part XII Diophantine triples using the Diophantine equation x² + D(∑ y_j²) = z² was used to produce four examples. Since the method did not allow a way to produce the 1,1,√2 triplet, an alternative but similar method had to be constructed to produce this particular triangle. The new equations not only produced the desired triangle but could be used to generate other ones as well.

Table I shows that the new equation for the x column now has a plus sign instead of a minus. Due to this change, the z, however, must be calculated by some other means.

Thus, in order to obtain the last column we have to resort to generating the squares, summing them up (columns 2 and 3) to obtain the appropriate z² value and then taking the square root of the z column. The result is the equation under the square root sign.

If we set D = 1 we get the equation a₁⁴ + 6n²a₁² + n⁴ having the roots a² = −3n² ± n²√8. Inspection of column four in Table III shows that the √8 is the expected result.

And so we have captured the elusive 1,1,√2 triangle. Finally in the last table we can generate more triples by increasing the number of a_js, or by increasing the values of D or n. Table IV depicts a pair of a_j groups along with a D = 2.

And, thus is generated the 11,√40, √161 triangle.

This concludes Part XIII. The next section (Part XIV) will show a novel way of obtaining sequences of fully integral Pythagorean triples.

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