Algebraic proof of non-trivial solution to the Pell's equation

Question

Let $d$ be a square-free positive integer, and consider the pairs $(x, y) \in \mathbb{Z}^2$ that satisfy:

$$x^2 - dy^2 = 1$$

The existence of a non-trivial solution to this equation (i.e. distinct from $(\pm 1, 0)$) is equivalent to showing existence of a torsion-free unit in $\mathbb{Z}[\sqrt{d}]$. All proofs I've seen of this fact uses the Dirichlet Approximation Theorem, and then follows by consecutive applications of the Pigeonhole Principle. But the group of units in $\mathbb{Z}[\sqrt{d}]$ is a purely algebraic construction, so I wonder if it's possible to prove this fact without resorting to the approximation theorem. Is there a known "algebraic" proof of this?

Answer 1

Given the Dirichlet Unit Theorem which states that the group of units in the ring of integers of a number field is finitely generated and has rank equal to $r = r_1 + r_2 − 1$, where $r_1$ is the number of real embeddings and $r_2$ half the number of complex embeddings of the field, you're guaranteed the existence of a fundamental unit in the ring of integers of a field $\mathbb{Q}[\sqrt{d}]$ as a real quadratic field has $r_1=2$. It's simple enough to translate back and forth between units and solutions of Pell's Equation as you've noted.

So to answer your question we would need to look at proof's of Dirichlet's Unit Theorem, and, in fact, most proofs apply Minkowski's Theorem to a logarithmic embedding of the units into a real vector space. Such proofs wouldn't qualify as "algebraic" proofs, but it is possible to give a more purely algebraic proof of Dirichlet's Unit Theorem by introducing the concept of ideles. Such an approach can be seen in Weil's Basic Number Theory or in Cassels and Frohlich Algebraic Number Theory where the unit theorem follows from the compactness of a certain idele group called $\mathbb{J}_K^1/K^\times$.

Answer 2

not sure what would satisfy you as "algebraic," however, Conway's topograph method, done with full detail (more than shown in his book) the way I have been drawing the diagram on this site, puts all sorts of numbers and labels on a drawing of $PSL_2 \mathbb Z.$ The construction of $u^2 - d v^2 = 1$ is incorporated into finding the matrix $$ P = \left( \begin{array}{rr} u & dv \\ v & u \end{array} \right) $$ which generates the (oriented) automorphism group of $$ G = \left( \begin{array}{rr} 1 & 0 \\ 0 & -d \end{array} \right), $$ that is $$ P^T G P = G. $$

The example in Hatcher is 19, $$ \left( \begin{array}{rr} 170 & 39 \\ 741 & 170 \end{array} \right) \left( \begin{array}{rr} 1 & 0 \\ 0 & -19 \end{array} \right) \left( \begin{array}{rr} 170 & 741 \\ 39 & 170 \end{array} \right) = \left( \begin{array}{rr} 1 & 0 \\ 0 & -19 \end{array} \right). $$ My scanner makes strange colors out of graph paper. That's life. The cycle of length six forms are the Gauss-Lagrange "reduced" forms equivalent to $x^2 - 19 y^2.$ I made a point of including the light blue numerical labels so that you could see where the reduced forms occur in the diagram. A form $\langle A,B,C \rangle$ is a pink value $A,$ a blue edge number $B$ (always positive, the little blue arrow indicates a sort of orientation) then a pink value $C,$ such that $AC < 0$ and $B > |A+C|.$ The fact that this description is equivalent to the usual definition of "reduced" is on page 37 of Franz's 2010 book, Theorem 1.36.

Let me put some links to my answers, then I will go look for publications, there is a new one by Hatcher, this was initiated in Conway, there was some helpful detail in Stillwell. See pages 70 and 71 in Hatcher for $d=19$ and the location of $P$ in the diagram. Oh, it appears that, after I had assumed he'd quit, Weissman is publishing his book, to be available in August 2017.

http://math.stackexchange.com/questions/81917/another-quadratic-diophantine-equation-how-do-i-proceed/144794#144794

http://math.stackexchange.com/questions/228356/how-to-find-solutions-of-x2-3y2-2/228405#228405

http://math.stackexchange.com/questions/342284/generate-solutions-of-quadratic-diophantine-equation/345128#345128

http://math.stackexchange.com/questions/487051/why-cant-the-alpertron-solve-this-pell-like-equation/487063#487063

http://math.stackexchange.com/questions/512621/finding-all-solutions-of-the-pell-type-equation-x2-5y2-4/512649#512649

http://math.stackexchange.com/questions/680972/if-m-n-in-mathbb-z-2-satisfies-3m2m-4n2n-then-m-n-is-a-perfect-square/686351#686351

http://math.stackexchange.com/questions/739752/how-to-solve-binary-form-ax2bxycy2-m-for-integer-and-rational-x-y/739765#739765 :::: 69 55

http://math.stackexchange.com/questions/742181/find-all-integer-solutions-for-the-equation-5x2-y2-4/756972#756972

http://math.stackexchange.com/questions/822503/positive-integer-n-such-that-2n1-3n1-are-both-perfect-squares/822517#822517

http://math.stackexchange.com/questions/1078450/maps-of-primitive-vectors-and-conways-river-has-anyone-built-this-in-sage/1078979#1078979

http://math.stackexchange.com/questions/1091310/infinitely-many-systems-of-23-consecutive-integers/1093382#1093382

http://math.stackexchange.com/questions/1132187/solve-the-following-equation-for-x-and-y/1132347#1132347 <1,-1,-1>

http://math.stackexchange.com/questions/1132799/finding-integers-of-the-form-3x2-xy-5y2-where-x-and-y-are-integers

http://math.stackexchange.com/questions/1221178/small-integral-representation-as-x2-2y2-in-pells-equation/1221280#1221280

http://math.stackexchange.com/questions/1404023/solving-the-equation-x2-7y2-3-over-integers/1404126#1404126

http://math.stackexchange.com/questions/1599211/solutions-to-diophantine-equations/1600010#1600010

http://math.stackexchange.com/questions/1667323/how-to-prove-that-the-roots-of-this-equation-are-integers/1667380#1667380

http://math.stackexchange.com/questions/1719280/does-the-pell-like-equation-x2-dy2-k-have-a-simple-recursion-like-x2-dy2

http://math.stackexchange.com/questions/1737385/if-d1-is-a-squarefree-integer-show-that-x2-dy2-c-gives-some-bounds-i/1737824#1737824 "seeds"

http://math.stackexchange.com/questions/1772594/find-all-natural-numbers-n-such-that-21n2-20-is-a-perfect-square/1773319#1773319

Answer 3

Assume that $d = p \equiv 1 \bmod 4$ is a prime number. The field of $p$-th roots of unity contains the real quadratic number field $k = {\mathbb Q}(\sqrt{p})$. The norm of the cyclotomic unit $1 + \zeta$ is a unit in $k$, and you will have an algebraic proof of the solvability of the Pell equation if you can show that this unit is not $\pm 1$. This must be east to prove using the sine function (this proof is due to Dirichlet).

If you want to avoid transcendental function, you will have to invoke congruences, but I haven't checked how far this will get you. Things will get more technical in general; for primes $p \equiv 3 \bmod 4$ you already have to look at the field of $4p$-th roots of unity.