I came across the following simple question: what odd integer squares have exactly 3 ones in their binary expansion?


After looking at it for a while I convinced myself that the only solutions to $r^2 = 1+ 2^m + 2^n$ (with $n > m \ge 3$) are $n=2m-2$ (the "trivial" case) and $(m,n) = (4,5),(4,9)$ (the "sporadic" cases). My first attempt was to analyze things 2-adically: write out the Taylor Series:
$f(x) = 1 - 2 \sum\_{n=1}^{\infty} (-1)^n C\_{n-1} x^n$, where $C\_n = \frac{1}{n+1} \binom{2n}{n}$ is the $n$-th Catalan number, and $v\_2(x) > 0$. We have $f(x)^2 = 1 + 4x$, so that $a\_m := \sqrt{1+2^m} = f(2^{m-2})$ (Note that $1+2^m$ is not a rational square when $m \ne 3$). If such an integer $r$ exists, then we must have $r \equiv \epsilon(a\_m + 2^{n-1}) \bmod {2^{n+1}}$, where $\epsilon = \pm 1, \pm (1+2^n)$ is one of the 4 square roots of 1 modulo $2^{n+1}$. Also, we must have $r < 2^{(n+1)/2}$. To get rid of the annoying cases, take all of this modulo $2^{n-1}$, where we find that the top $(n-3)/2$ bits of $a\_m \bmod 2^{n-1}$ must be either all 0's or all 1's. By writing out the first few terms of $f(x)$ I found that there's a run of $m-2$ 0's which corresponds to the trivial solution, and a run of $m-1$ 1's which corresponds to the sporadic solution (the condition that $m-1 \ge (n-3)/2$ limits the possible $m$'s and $n$'s to a small set). However, I don't see any easy way to show that there are no long "runs" of 1's or 0's among the higher bits of $a\_m$ (by the bits, I mean to write out $a\_m = \sum\_{n=0}^{\infty} b\_{m,n} 2^n$, where $b\_{m,n} \in \{0,1\}$). Such a statement would be to the effect that we can't approximate $a\_m$ too closely by small integers -- this is the sort of $p$-adic Diophantine approximation statement that I would think should be true, but I can't find it.


Absent that, another approach is via $S$-unit equations. This show (see below) that for each $m$, there are only a finite number $n$'s for which $1+2^m + 2^n$ is a rational square, but I'd like to prove (with the exception of the sporadic solution) that there is exactly 1. Since, as I mentioned above, $1+2^m$ is not a rational square when $m > 3$, denote by $K = K\_m = \mathbb{Q}(\sqrt{1+2^m})$. It's easy to see that the prime 2 splits in $K$: call $\frak{a},\frak{a}'$ the ideals above 2. Let $k>0$ denote the order of $\frak{a}$ in the ideal class group of $K$ and $\beta$ a generator of the principal ideal $\frak{a}^k$. Also denote conjugation of $K/\mathbb{Q}$ by $\quad '$. Then we have $r+\sqrt{1+2^m} = \epsilon \beta^t 2^u$, for some non-negative integers $t,u$ and $\epsilon \in K$ is a unit. Note that we might need to look at $r-\sqrt{1+2^m}$ instead. By taking norms we see that $2^{2n} = 2^{2u + kt}$, so that $2n = 2u+kt$. By noting that $\epsilon \beta^t 2^u - \epsilon' {\beta'}^t 2^u = 2 \sqrt{1+2^m}$, we see that $u=0,1$. For each of those values we then have an equation of the form $\alpha - \alpha' = \gamma$, where $\gamma$ is fixed, and $\alpha,\alpha'$ are $S$-units (here $S$ consists of the primes above 2), so we know that there are only a finite number of solutions. Any suggestions as to how to proceed to show that my initial guesses are correct?



Binary expansion of squares
This was solved in a paper of Szalay (in Indag. Math. in 2002), using lower bounds for approximations to $\sqrt{2}$ by rationals with denominators of the form $2^k$ (obtained by Beukers using Pad\'e approximations to $\sqrt{1-z}$).


