On the Compatibility of Bézout Coefficients between Pythagorean Pairs under Unimodular Transformations

In a recent preprint, Gullerud and Walker proved a theorem and made a conjecture about thecorrectness of efficiently generating Bézout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, conrm that their conjecture is true, and furthermore we give a generalization.

From integer Lorentz transformations to Pythagoras

In this article we show how to obtain by recurrence all primitive Pythagorean triples from the single basic (4, 3, 5) triple. It is all done by repeated application of two integer unimodular transformations that leave the indefinite metric x2 + y2 - z2 invariant, together with the additional transformations (i) that change the sign of x and (ii) that change the sign of y. These alone would restrict the triples to those in which x is even, y is odd and z is positive, so we then include two further transformations (iii) that exchange x and y and (iv) that change the sign of z, thereby accounting for all primitive triples. There is a bonus in extending the method beyond the first objective in that the theory, which we outline, of the Lorentz transformations involved not only provides necessary background explanation of the first part, but it also enables us to show how to solve by recurrence Diophantine equations of the form x2 + y2 = z2 + n, where n is any integer.