---

Claim. For a positive integer $M$ let $R_M$ be the set of positive integers congruent to $0$, $1$ or $2$ modulo $M$, and put

$$A=\bigcup_{k\ge0}\left( \left(R_{2q}\cap\left[\frac{4q^2}{9},4q^2\right]\right) \cup \left(R_q\cap\left[\frac{5q^2}{9},q^2\right]\right) \right),\qquad q=3^k.$$

Every positive integer is $m^2+a$ with an integer $m\ge0$ and $a\in A$, and, writing $A(X)=\lvert A\cap[1,X]\rvert$ for real $X$,

$$\liminf_{X\to\infty}\frac{A(X)}{\sqrt X}=3\sqrt2, \qquad \limsup_{X\to\infty}\frac{A(X)}{\sqrt X}=\frac92.$$

The upper limit is witnessed along $X=3^{2k}$. Hence the smallest possible value of the limsup in the first question of Problem 33 is at most $9/2$.

Covers. An upper bound of $9/2$ for the smallest possible limsup, the first question of Problem 33. The value lies below the pending upper bounds of van Doorn, $2\varphi^{5/2}\approx6.66$, and of the unreviewed write-up of 2026-09-07, $13\sqrt2/4\approx4.596$. The claim does not determine the smallest limsup and asserts neither optimality of this set nor a bound $A(X)\le(9/2)\sqrt X$ at every cutoff. The lower limit $3\sqrt2$ is that of this one set and says nothing about the liminf question, which concerns every complement.

Standing. L3 is author-recorded at tier 0: a complete symbolic proof, including a uniform counting error at every real cutoff, without independent examination or a Lean proof. Catalog acceptance is pending.

Depends on. L3 (tier 0), which consumes no other native claim and no external theorem.