Erdős Problem #1081

Erdős Problem 1081: counting sums of two squarefull numbers

1. Statement and known answer

Let $A(x)$ count the positive integers at most $x$ expressible as the sum of two positive squarefull numbers. A positive integer is squarefull if every prime dividing it has exponent at least two. The proposed asymptotic $A(x)\sim cx/\sqrt{\log x}$ is false. Blomer and Granville's Corollary 2 in Estimates for representation numbers of quadratic forms, Duke Math. J. 135 $2006$, gives constants $B_1,B_2$ such that \[ \frac{x(\log\log x)^{B_1}}{(\log x)^\alpha}\ll A(x)\ll \frac{x(\log\log x)^{B_2}}{(\log x)^\alpha},\qquad \alpha=1-2^{-1/3}<\frac12. \] Their primary text is https://dms.umontreal.ca/~andrew/PDF/quadraticforms.pdf; the source problem is https://www.erdosproblems.com/1081. The lower bound makes $A(x)/(x/\sqrt{\log x})$ tend to infinity, since a positive power of $\log x$ dominates any fixed power of $\log\log x$. That deep distinct-value estimate is background, not a proof supplied by this attempt. Below we prove an exact parametrization and an asymptotic for the number of representations, then identify why these do not yet reconstruct the disproof.

2. Counting squarefull integers

Every squarefull integer has a unique representation $a^2b^3$ with $a\ge1$ and $b\ge1$ squarefree. At a prime, an even exponent is $2u$; an odd exponent, necessarily at least three, is $2u+3$. Choosing whether the prime divides $b$ according to the exponent's parity gives existence and uniqueness. Writing $Q(y)$ for the number of positive squarefull integers at most $y$, we obtain \[ Q(y)=\sum_{b\le y^{1/3}}\mu^2(b)\left\lfloor\frac{\sqrt y}{b^{3/2}}\right\rfloor =C\sqrt y+O(y^{1/3}),\qquad C=\sum_{b\ge1}\frac{\mu^2(b)}{b^{3/2}}=\frac{\zeta(3/2)}{\zeta(3)}. \] The floor errors total $O(y^{1/3})$, while the omitted tail is bounded by $\sqrt y\sum_{b>y^{1/3}}b^{-3/2}=O(y^{1/3})$. The Euler product identity for $C$ follows from $1+p^{-3/2}=(1-p^{-3})/(1-p^{-3/2})$. This also proves $Q(y)=O(\sqrt y)$ for $y\ge1$.

3. An asymptotic for ordered representations

Let $R(x)$ count ordered pairs of positive squarefull integers $(u,v)$ with $u+v\le x$, counting every representation separately. For real $x\ge2$, \[ R(x)=\sum_{\substack{u\le x\\u\text{ squarefull}}}Q(x-u) =C\sum_{\substack{u\le x\\u\text{ squarefull}}}\sqrt{x-u}+O(x^{5/6}). \] For $0\le y<1$ the estimate $Q(y)=C\sqrt y+O(y^{1/3})$ still holds after enlarging its constant; at zero both sides vanish. Thus the error bound follows by bounding each summand by $O(x^{1/3})$ and using $Q(x)=O(\sqrt x)$. Stieltjes integration by parts gives \[ \sum_{u\le x,\ u\text{ squarefull}}\sqrt{x-u} =\frac12\int_0^x\frac{Q(t)}{\sqrt{x-t}}\,dt =\frac C2\int_0^x\frac{\sqrt t}{\sqrt{x-t}}\,dt+O(x^{5/6}). \] One may first integrate to $x-h$ and let $h\downarrow0$; the endpoint term tends to zero and the displayed integrals converge. Substitution $t=x\sin^2\theta$ evaluates the main integral as $\pi x/2$. The error integral is $O(x^{5/6})$ by $t=xs$. Hence \[ \boxed{R(x)=\frac{\pi C^2}{4}x+O(x^{5/6}).} \] This is a representation-count theorem, not an asymptotic for $A(x)$.

4. Distinct values and the remaining obstruction

The parametrization makes the support being counted by $A(x)$ the union of the values \[ b^3a^2+d^3c^2\le x,\qquad a,c\ge1,\quad b,d\ge1\text{ squarefree}. \] These binary quadratic forms overlap, and a single value may have many representations. Even knowing their total number to an $o(x)$ error does not control the size of this union. For a baseline, the subfamily $b=d=1$ contains all sums of two positive squares. The classical Landau--Ramanujan theorem, if used as a stated external input, gives $A(x)\gg x/\sqrt{\log x}$: excluding representations with a zero square removes at most $O(\sqrt x)$ values. This reaches the originally guessed scale but does not exceed it. The missing step is a sufficiently uniform lower bound for the distinct union over a growing collection of squarefree parameters, including control of overlaps. Blomer--Granville address this through representation numbers and varying discriminants; their Corollary 2 resolves it, but that analytic argument is not reproduced above. It would be invalid to replace $A(x)$ by $R(x)$ or to infer the disproof solely from the boxed formula.

5. Assessment

The squarefull counting formula, its error term, the quadratic-form reduction and the ordered-representation asymptotic are proved. The known negative answer is correctly identified and its implication explained. A reconstruction of the distinct-value lower bound needed for the negative answer remains unresolved in this attempt.

Completion rate estimate: 55%.