Prove that $0.a_1b_1a_2b_2a_3b_3 \ldots = 0.21742615\ldots$ is irrational, where $(a_i,b_i)$ are the solutions of $x^2-3y^2=1$

Question

Let $S$ be the set of all ordered pairs of positive integers $(x,y)$ such that $x^2-3y^2 = 1$. Let $(a_1,b_1),(a_2,b_2),\ldots$ be the elements of $S$ listed in ascending order, i.e., $a_i < a_{i+1}$ for all $i \geq 1$. Prove that $$k = 0.a_1b_1a_2b_2a_3b_3 \ldots = 0.21742615\ldots$$ is irrational.

The solutions to the equation $x^2-3y^2 = 1$ are the solutions to $x+y\sqrt{3} = (2+\sqrt{3})^m$. Therefore, $$x = \sum_{i=0}^{\left\lfloor\frac{m}{2}\right\rfloor} 2^{m-2i} \cdot 3^i \binom{m}{2i} \quad \text{and} \quad y = \sum_{i=0}^{\left\lfloor\frac{m-1}{2}\right\rfloor}2^{m-2i-1} \cdot 3^i\binom{m}{2i+1}.$$ How can we use this to find the value of $k$?

Answer 1

Hints: If $k$ is rational then its decimal expansion repeats after some point. Let $p$ be the period, and let $N=10^p$. Then there is a finite set $F$ of pairs such that $$ a_i,b_i\in\{(r+sN^n)/(N-1)\mid(r,s)\in F,\,n\geq0,\,s>0\} $$ for $i$ sufficiently large. For some fixed $(r,s),(t,u)\in F$ with $s,u>0$, there are infinitely many $n$ for which $$ \left(\frac{r+sN^n}{N-1}\right)^2-3\left(\frac{t+uN^n}{N-1}\right)^2=1. $$ Now expand: $$ (r^2-3t^2)+2(rs-3tu)N^n+(s^2-3u^2)N^{2n}=(N-1)^2 $$ and equate coefficients of powers of $N^n$: $$ r^2-3t^2=(N-1)^2,\hspace{10mm}rs-3tu=s^2-3u^2=0. $$ Then $$ s^2(N-1)^2=s^2r^2-3s^2t^2=9t^2u^2-9t^2u^2=0, $$ a contradiction.