For a squarefree integer $m\geq 1$ and an integer $n$, define $ T_m(n)={\,k\geq 1:\ n-2^k = m u^2 \text{ for some } u\in \mathbb Z_{\geq 0}\,}. $

So $T_m(n)$ records the exponents $k$ for which the dyadic shift $n-2^k$ has squarefree part equal to $m$.

In this note I prove that for every odd squarefree $m\geq 3$, $ |T_m(n)|\leq 2. $

For the pure square case $m=1$, I prove the sharp bound $ |T_1(n)|\leq 3. $

Moreover, equality holds infinitely often. For example, for every $t\geq 1$, $ 17\cdot 4^t=(2^{t+2})^2+2^{2t}=(3\cdot 2^t)^2+2^{2t+3}=(2^t)^2+2^{2t+4}, $ so $ |T_1(17\cdot 4^t)|=3. $

The argument is elementary. In the odd squarefree case, one compares two adjacent identities $ n-2^{k_i}=m u_i^2,\qquad n-2^{k_{i+1}}=m u_{i+1}^2, $ subtracts them, and analyzes the resulting factorization $ m(u_i-u_{i+1})(u_i+u_{i+1})=2^{k_i}(2^{k_{i+1}-k_i}-1). $ This gives enough rigidity to rule out three occurrences of the same squarefree kernel.

As a corollary, along the sequence $ n-2,\ n-4,\ n-8,\ n-16,\dots $ the squarefree kernel cannot repeat too often. In particular, among the first $K$ shifts there are at least about $K/2$ distinct squarefree kernels.

This is a very small note, but I think the statements are neat and perhaps not completely obvious at first sight.

Here is the note.