An SO(2,2) Iterated Function System Part 5 - Back to July 1

 Researchers believe that cats have an abstract understanding of numbers, often up to about seven. Some claim that mother cats can count as high as six or seven, though three or four is more likely. In contrast, humans seem able to count much further. George Cantor, for example, explored numbers reaching to infinity—and even beyond. Personally, I find numbers challenging, both in practice and theory. Number theory has always been a painful experience for me. 

Still, when there’s a need, there’s always a way. So, in this blog post, we draw a final dot to mark the end of our journey through the ring of integer.

Let us start with a quote from July 1 post "Tuesday Special - Tetractys and Lattice Infinity":

 I could not find anything about tetrads in Babylonia, but I found them on math.stackexchange: Diophantine equation a² + b² =c² + d². The complete solution can be found in the textbook L.J. Mordell, "Diophantine Equations", Academic Press 1969, on p. 15.

Well, it is not explicitly complete there, it is somewhat sketchy, but here it is (I skip the proof).

Proposition 1. Every primitive solution of  (1) is of the form

a = (mp+nq)/2,
b = (np-mq)/2,
c = (mp-nq)/2,
d = (mq+np)/2,

where m,n,p,q are integers. Conversely, for any integers m,n,p,q such that a,b,c,d are integers, the formula above provides a solution of  a² + b² =c² + d².

There I quoted a Proposition from Mordell's book, without a proof. But now, with Proposition 1 from the previous post,  we have a complete proof of Mordell's statement. Let us discuss this in details.

Suppose a,b,c,d are integers satisfying a² + b²  = c² + d². Let x be a vector in R^2,2 with components (a,b,c,d). Then x is a null vector: Q(x) =  a² + b²  - c² - d² = 0. The matrix x^, defined as in  Part 2, is

x^ = {{c+a,b+d},{b-d,c-a}}                (1)

is, automatically, of determinant zero, with integer  components. Thus we can apply Proposition 1 from Part 4 to deduce that there are vectors v,w with integer components such that

x^ = vw^T.                 (2)

Let (p,-q) be the components of v, and let (m,n) be the components of w. Then (2) takes the form

{{c+a,b+d},{b-d,c-a} = {{pm,pn},{-qm,-qn}}            (3).

From (3) we immediately get

a = (pm+qn)/2,

b= (pn-qm)/2,

c = (pm-qn)/2,

d = (pn+qm)/2,

which are exactly the formulas from Proposition 1 in Mordell's text. In order to get the exact correspondence we have set (p,-q) to be the components of v, but that does not really matter, since if q runs through all integers, so does -q.