## JSH: Physics relevance of Pell's Equation?

I'm curious about how much this bit of number theory comes up in physics.

Mostly I see things on zeroes of Racah coefficients. Here's one:

W. A. Beyer1, J. D. Louck1 and P. R. Stein1

(1) Los Alamos National Laboratory, 87545 Los Alamos, NM, U.S.A.
Received: 11 February 1986 Revised: 22 August 1986

Abstract The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation. The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.

I'm curious about how big of a deal it is to know an easier way to solve them, as I've noted the following:

Given x^2 - Dy^2 = 1, for instance, if you have a solution to the negative Pell's Equation,

j^2 - Dk^2 = -1

then you have a solution for x, from x = 2j^2 + 1.

For instance, 16 - 17 = 1, so j=4, k=1, and I have x=2(16) + 1=33.

And of course, 33^2 - 17*8^2 = 1.

It's rather trivial number theory to show that all positive integer values for D can be handled by alternates like the negative Pell's Equation, where you get the square improvement with all.

There is a bit of a puzzle I think as to why the better way to approach Pell's Equation is not commonly taught.

Unless someone comes up with a reference that mentions it, so far what I've seen on-line is just a direct tackling of Pell's Equation itself using various methodologies, which then, is mathematically naive (sorry but that is I think the best description).

I have been pushing this issue for a few days, but now I'm more curious about possible explanations different from what I have.

So then, what is the physics usefulness of Pell's Equation? Is the zeroes of the Racah coefficient it?

Do techniques for solving it already somehow encompass the simple result I've found?

If not, what then? How can the literature be updated?

IF so, where is that shown?