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\title{Erd\H{o}s Problem \#1054: a consolidated write-up of the forum proofs}
\author{Compiled from comments by\\[2pt] jif, Terence Tao, Vjekoslav Kova\v{c}, Thomas Bloom,\\ Liam Price/GPT-5.5 Pro, and principia\_math}
\date{June 24, 2026}

\begin{document}
\maketitle

\begin{abstract}
This note consolidates the main proof ideas appearing in the discussion thread for Erd\H{o}s Problem \#1054.  The problem asks about the least integer $m$ for which $N$ is the sum of the $k$ smallest divisors of $m$, for some $k\geq 1$.  The write-up has three purposes.  First, it records jif's proof/certification that the function is defined for every positive integer except $2$ and $5$.  Second, it gives a clean version of the Tao--Kova\v{c} density argument, culminating in Kova\v{c}'s strengthened bound
\[
 \#\{N\leq X:f(N)\leq \delta N\}\ll \exp\{-\exp((1/\delta)^c)\}X
\]
for some $c>0$ and sufficiently small $\delta$.  Third, it records, but does not reproduce, the June 2026 principia\_math/Principia Math claim resolving the remaining limsup question, because the forum page gives only a summary and links to an external Overleaf/Lean proof.  All attributions are made to the forum names or author names used in the thread.
\end{abstract}

\section{Problem, notation, and attributions}

Let
\[
 1=d_1(m)<d_2(m)<\cdots<d_{\tau(m)}(m)=m
\]
be the positive divisors of $m$ in increasing order.  We say that $N$ is represented by $m$ if
\[
  N=d_1(m)+\cdots+d_k(m)
\]
for some $1\leq k\leq \tau(m)$.  Erd\H{o}s Problem \#1054 asks about
\[
 f(N)=\min\{m\geq 1:N\hbox{ is represented by }m\},
\]
when this minimum exists.  The question, as stated on the Erd\H{o}s Problems site and attributed there to Guy's collection, is whether $f(N)=o(N)$, perhaps for almost all $N$, and whether
\[
  \limsup_{N\to\infty}\frac{f(N)}{N}=\infty.
\]
The formulation is closely related to Erd\H{o}s Problem \#468, with an index shift coming from whether $1$ is counted among the divisors.

For density estimates it is convenient to put $f(2)=f(5)=\infty$, since these two values are not represented; this convention does not affect any asymptotic statement.

For $j\geq0$ define
\[
  \sigma_j(m)=\sum_{i=1}^{\tau(m)-j}d_i(m),
\]
with the convention that the sum is $0$ if $j\geq \tau(m)$.  Thus $\sigma_j(m)$ is the sum of all divisors of $m$ except the $j$ largest ones.  The condition ``$N$ is the sum of the $k$ smallest divisors of $m$'' is the same as
\[
  N=\sigma_j(m) \qquad (j=\tau(m)-k).
\]

The principal contributions used below are as follows.
\begin{itemize}[leftmargin=2.2em]
\item \textbf{jif} gave a proof/certificate that the represented positive integers are exactly $\N\setminus\{2,5\}$, using finite verification, prime-window subset-sum intervals, Rosser--Schoenfeld bounds, and an explicit consequence of Helfgott's ternary Goldbach theorem.
\item \textbf{Terence Tao} gave the original density obstruction to $f(N)=o(N)$, via estimates for the sums $\sigma_j(m)$ and a Markov/union-bound argument.
\item \textbf{Vjekoslav Kova\v{c} (Vjeko\_Kovac)} wrote a short self-contained note, based on Tao's argument and comments by Thomas Bloom, proving a much stronger uniform upper bound for the density of $\{N:f(N)\leq\delta N\}$.
\item \textbf{Thomas Bloom} pointed out the dependence on the moment parameter and how to optimize it.
\item \textbf{Liam Price} posted links to GPT-5.5 Pro proofs giving intermediate fixed-power improvements.  These are superseded by Kova\v{c}'s shorter argument, but are part of the thread history.
\item \textbf{principia\_math / Principia Math} posted a claimed resolution of the remaining limsup question.  Since the thread text gives only the statement and links to an Overleaf/Lean proof, this note records that claim separately rather than treating it as independently checked here.
\end{itemize}

\section{Representability and well-definedness}

We first isolate the elementary obstruction.

\begin{proposition}[The exceptional values $2$ and $5$]
The positive integers $2$ and $5$ are not represented.  The integers $1,3,4$ are represented.
\end{proposition}

\begin{proof}
Every nonempty prefix sum of divisors begins with $1$.  Hence a two-term prefix sum is at least $1+2=3$, so $2$ cannot occur.  If $5$ were a two-term prefix sum, it would have to be $1+4$, but $4$ cannot be the smallest divisor after $1$, since a number whose least nontrivial divisor is $4$ would have no divisor $2$ and hence would be odd, impossible if it is divisible by $4$.  With three or more terms the prefix sum is at least $1+2+3=6$ when it exists; in any case it is never $5$.  Finally $1$, $3$, and $4$ are represented by $m=1,2,3$, respectively.
\end{proof}

The positive result rests on two standard gadgets plus explicit finite certificates.

\begin{lemma}[Prime-window prefix gadget, jif]
Let $M<p_1<\cdots<p_t\leq X<M^2$ be primes, and let
\[
  m=p_1p_2\cdots p_t.
\]
Then the first divisors of $m$ are
\[
  1,p_1,p_2,\ldots,p_t.
\]
Consequently, if $N=p_{i_1}+\cdots+p_{i_s}$ is a subset sum of primes in $(M,X]$, then $N+1$ is represented by $m=p_{i_1}\cdots p_{i_s}$.
\end{lemma}

\begin{proof}
Every composite divisor of $m$ is a product of at least two primes, hence is $>M^2$.  Since $X<M^2$, all primes $p_i\leq X$ occur before any composite divisor.  Taking the product over a selected subset gives initial divisors $1$ followed by exactly the selected primes in increasing order, and the relevant prefix sum is $1+N$.
\end{proof}

\begin{lemma}[Interval-extension trick, jif]
Suppose subset sums of already chosen positive integers cover every integer in an interval $[C,U]$.  If the next available integer $p$ satisfies
\[
  p\leq U-C+1,
\]
then after adjoining $p$ the subset sums cover $[C,U+p]$.
\end{lemma}

\begin{proof}
The old sums cover $[C,U]$ and the old sums plus $p$ cover $[C+p,U+p]$.  The hypothesis $p\leq U-C+1$ says $C+p\leq U+1$, so these two integer intervals touch or overlap.  Their union is therefore $[C,U+p]$.
\end{proof}

The following theorem is jif's well-definedness result as recorded in the forum and in the accompanying GitHub repository.  The hand proof below identifies the elementary structure; the finite and numerical inequalities are those certified in the repository's verifier output.

\begin{theorem}[jif's representability theorem]
Assuming the finite certificates and numerical inequalities verified in jif's repository, every positive integer other than $2$ and $5$ is represented.  Equivalently, $f(N)$ is finite exactly for
\[
  N\in\N\setminus\{2,5\}.
\]
\end{theorem}

\begin{proof}
The values $1,3,4$ are represented and $2,5$ are not, by Proposition 2.1.  It remains to cover $n\geq6$.

The verifier records four overlapping ranges.
\begin{enumerate}[label=(\roman*),leftmargin=2.4em]
\item A direct divisor-prefix search covers every $6\leq n\leq 10^7$, except that the value $7$ is separately represented as $7=1+2+4$ by $m=4$.
\item A prime-window certificate with $M=10^4$ and $X=99,000,000<M^2$ gives representations for every
\[
 469,616\leq n\leq 273,803,744,799,154.
\]
\item A larger seed proves that every integer in $[105,000,000,156,000,000]$ is a sum of distinct primes in $(20,000,000,40,000,000)$.  Since the seed length is $51,000,001$ and the next prime after $40,000,000$ is $40,000,003$, the interval-extension lemma starts.  Bertrand's postulate continues the induction, and Rosser--Schoenfeld bounds for $\pi(x)$ lower-bound the available prime mass up to $399,000,000,000,000<20,000,000^2$.  This yields representations for every
\[
 105,000,001\leq n\leq 1,111,351,202,532,220,892,436,000,001.
\]
\item For the tail one uses the explicit Helfgott-type input certified by jif: every odd $T\geq10^{27}$ is a sum of three distinct odd primes, each larger than $T/(30000\log T)$.  The numerical margins excluding small or repeated coordinates are checked using the explicit constants quoted from Helfgott's Section 7 and Rosser--Schoenfeld's estimate for sums of $\Lambda$.
\end{enumerate}

We spell out how the tail input yields divisor-prefix representations.  Let $n$ be even and large enough that $T=n-1\geq10^{27}$.  Write
\[
  n-1=p+q+r
\]
with $p,q,r$ distinct odd primes, each $>T/(30000\log T)$.  For $T\geq10^{27}$ this lower bound gives $pq>T>r$.  Hence for $m=pqr$ the initial divisors are $1,p,q,r$, and their sum is $n$.

For odd $n\geq10^{27}+10^8$, choose a prime
\[
  60000\log n<\ell<120000\log n
\]
using Bertrand's postulate.  Put $T=n-1-\ell$, which is odd and at least $10^{27}$ in the range under consideration.  Write $T=p+q+r$ as above.  The verifier checks the elementary slack inequalities
\[
  \ell<p, \qquad \ell p>r.
\]
Therefore the initial divisors of $m=\ell pqr$ are $1,\ell,p,q,r$, and their sum is $n$.

The four covered intervals
\[
[6,10^7],\quad
[469,616,273,803,744,799,154],\quad
[105,000,001,1,111,351,202,532,220,892,436,000,001],
\]
\[
[10^{27}+10^8,\infty)
\]
overlap.  Thus every $n\geq6$ is represented.  Together with Proposition 2.1 this proves the claim, conditional only on the stated verifier outputs and explicit analytic inputs.
\end{proof}

\begin{remark}
This section is deliberately phrased as a certificate-based proof.  The arithmetic for the large finite intervals is not reproduced line by line in this note; it is part of jif's Rust verifier and committed output.
\end{remark}

\section{The density obstruction of Tao and its sharpening by Kova\v{c}}

The decisive observation is that representations with $m\leq \delta N$ force a large value of $\sigma_j(m)/m$.

Let
\[
 E_\delta(X)=\{N\leq X:f(N)\leq \delta N\}.
\]
If $N\in E_\delta(X)$ and $m=f(N)$, then $m\leq \delta X$ and, for some $j\geq0$,
\[
 N=\sigma_j(m),\qquad \frac{\sigma_j(m)}{m}=\frac{N}{m}\geq \frac1\delta.
\]
Thus estimates for the moments of $\sigma_j(m)/m$ immediately give density bounds for $E_\delta(X)$.

\subsection{Kova\v{c}'s moment estimate}

The following is the core estimate in Kova\v{c}'s note.  It is a cleaned-up version of his proof, based on Tao's original idea and Bloom's comments on the dependence on the moment parameter.

\begin{lemma}[Kova\v{c}]
There is an absolute constant $C>0$ such that, for every integer $q\geq3$ and every real $x\geq1$,
\[
 \sum_{m\leq x}\sum_{j\geq0}\left(\frac{\sigma_j(m)}{m}\right)^q
 \ll \exp(Cq\log\log q)x.
\]
The implicit constant is absolute.
\end{lemma}

\begin{proof}
Let
\[
  1=r_1<r_2<\cdots<r_{\tau(m)}=m
\]
be the divisors of $m$.  Reflecting the divisor list gives
\[
  \frac{\sigma_j(m)}m=\sum_{i>j}\frac1{r_i}.
\]
After expanding the $q$-th power and then summing over $j$, a fixed tuple $a_1,\ldots,a_q$ of divisors is counted once for each divisor threshold not exceeding $\min a_i$.  Hence
\[
 \sum_{j\geq0}\left(\frac{\sigma_j(m)}m\right)^q
 =
 \sum_{\substack{r,a_1,\ldots,a_q\mid m\\ r\leq \min(a_1,\ldots,a_q)}}
 \frac1{a_1\cdots a_q}.
\]
Summing over $m\leq x$ and bounding the number of multiples of $[r,a_1,\ldots,a_q]$ by $x/[r,a_1,\ldots,a_q]$, it is enough to prove
\[
 S:=\sum_{\substack{r,a_1,\ldots,a_q\geq1\\ r\leq \min a_i}}
 \frac1{a_1\cdots a_q [r,a_1,\ldots,a_q]}
 \ll \exp(Cq\log\log q).
\]
For each $i$ put
\[
 b_i=\frac{r}{(r,a_i)},\qquad c_i=\frac{a_i}{(r,a_i)}.
\]
Then $b_i\mid r$, $(b_i,c_i)=1$, $a_i=rc_i/b_i$, and
\[
 [r,a_1,\ldots,a_q]=r[c_1,\ldots,c_q].
\]
The condition $r\leq a_i$ becomes $b_i\leq c_i$.  Dropping the coprimality condition only increases the sum, so
\[
 S\leq
 \sum_{r\geq1}\frac1{r^{q+1} }
 \sum_{b_1,\ldots,b_q\mid r}b_1\cdots b_q
 \sum_{c_i\geq b_i}
 \frac1{c_1\cdots c_q[c_1,\ldots,c_q]}.
\]
Fix $0<\beta<1$.  Since $c_i\geq b_i$,
\[
 \frac1{c_i}\leq \frac1{b_i^{1-\beta}c_i^\beta}.
\]
Therefore
\[
 S\leq S'S'',
\]
where, with $\beta=(q-1)/q$,
\[
 S'=\sum_{r\geq1}\frac1{r^2}
 \left(\sum_{d\mid r}\frac1{d^{(q-1)/q}}\right)^q
\]
and
\[
 S''=\sum_{c_1,\ldots,c_q\geq1}
 \frac1{c_1^{(q-1)/q}\cdots c_q^{(q-1)/q}[c_1,\ldots,c_q]}.
\]
It remains to estimate these two Euler products.

For $S'$, multiplicativity gives
\[
 S'=\prod_p L_p,
\]
where
\[
 L_p=\sum_{\ell\geq0}p^{-2\ell}
 \left(\sum_{0\leq \gamma\leq \ell}p^{-((q-1)/q)\gamma}\right)^q.
\]
For $p\leq q$,
\[
 L_p\leq (1-p^{-2})^{-1}(1-p^{-(q-1)/q})^{-q}.
\]
Using
\[
 \sum_{p\leq q}p^{-(q-1)/q}
 \leq q^{1/q}\sum_{p\leq q}\frac1p
 \ll \log\log q,
\]
we obtain
\[
 \prod_{p\leq q}L_p\ll \exp(Cq\log\log q).
\]
For $p>q$, one has $q p^{-(q-1)/q}\ll1$ and hence
\[
 L_p\leq 1+O(p^{-2}).
\]
Thus $\prod_{p>q}L_p\ll1$, and so
\[
 S'\ll \exp(Cq\log\log q).
\]

For $S''$, again by multiplicativity,
\[
 S''=\prod_p M_p,
\]
where
\[
 M_p=\sum_{\gamma_1,\ldots,\gamma_q\geq0}
 p^{-((q-1)/q)(\gamma_1+\cdots+\gamma_q)-\max_i\gamma_i}.
\]
Let
\[
 A_h=\sum_{0\leq \gamma\leq h}p^{-((q-1)/q)\gamma}.
\]
Grouping terms by $h=\max_i\gamma_i$ gives
\[
 M_p=\sum_{h\geq0}p^{-h}(A_h^q-A_{h-1}^q),\qquad A_{-1}=0.
\]
For $p\leq q$ the rough bound
\[
 A_h\leq (1-p^{-(q-1)/q})^{-1}
\]
yields
\[
 M_p\leq (1-p^{-1})^{-1}(1-p^{-(q-1)/q})^{-q},
\]
and therefore
\[
 \prod_{p\leq q}M_p\ll \exp(Cq\log\log q).
\]
For $p>q$, the mean value theorem gives
\[
 A_h^q-A_{h-1}^q\ll q p^{-h(q-1)/q},
\]
so
\[
 M_p-1\ll q\sum_{h\geq1}p^{-h(1+(q-1)/q)}
 \ll q p^{-2+1/q}.
\]
Consequently
\[
 \sum_{p>q}(M_p-1)
 \ll q\sum_{n>q}n^{-2+1/q}\ll1,
\]
and $\prod_{p>q}M_p\ll1$.  Hence
\[
 S''\ll \exp(Cq\log\log q).
\]
Combining the estimates for $S'$ and $S''$ proves the lemma, after adjusting $C$.
\end{proof}

\subsection{The strengthened density theorem}

\begin{theorem}[Kova\v{c}; strengthening Tao's obstruction]
There exists $c>0$ such that, for all sufficiently small $\delta>0$ and all $X\geq1$,
\[
 \#\{N\leq X:f(N)\leq \delta N\}
 \ll \exp\{-\exp((1/\delta)^c)\}X.
\]
Consequently, for every fixed $M>0$,
\[
 \#\{N\leq X:f(N)\leq \delta N\}
 \ll_M \delta^M X
\]
uniformly in $X$.
\end{theorem}

\begin{proof}
Let $E_\delta(X)$ denote the set on the left.  If $N\in E_\delta(X)$ and $m=f(N)$, then $m\leq \delta X$ and $N=\sigma_j(m)$ for some $j\geq0$ with $\sigma_j(m)/m\geq1/\delta$.  Therefore, for every integer $q\geq3$,
\begin{align*}
 \#E_\delta(X)
 &\leq
 \#\left\{(m,j):m\leq \delta X,\ \frac{\sigma_j(m)}m\geq\frac1\delta\right\}  \\
 &\leq
 \delta^q\sum_{m\leq \delta X}\sum_{j\geq0}
 \left(\frac{\sigma_j(m)}m\right)^q  \\
 &\ll
 \delta^{q+1}\exp(Cq\log\log q)X,
\end{align*}
where Lemma 3.1 is used in the last line.

Choose $c=1/(2C)$, after enlarging $C$ if necessary, and take
\[
 q=\left\lfloor\exp((1/\delta)^c)\right\rfloor.
\]
Then
\[
 \delta^{q+1}\exp(Cq\log\log q)
 =\exp\left(-(q+1)\log\frac1\delta+Cq\log\log q\right).
\]
Because $\log\log q\sim c\log(1/\delta)$, the exponent is at most
\[
 -\frac12 q\log\frac1\delta
\]
for sufficiently small $\delta$.  This is stronger than
\[
 -\exp((1/\delta)^c)
\]
after slightly reducing $c$, and the theorem follows.  The fixed-power bound $O_M(\delta^M X)$ is immediate from the double-exponential decay as $\delta\to0$.
\end{proof}

\begin{corollary}[Failure of the ``almost all'' $o(N)$ statement]
For all sufficiently small fixed $\delta>0$, the set
\[
 \{N\in\N:f(N)>\delta N\}
\]
has lower density arbitrarily close to $1$.  In particular, $f(N)=o(N)$ is false even as an almost-all statement.
\end{corollary}

\begin{proof}
Theorem 3.2 gives
\[
 \sup_{X\geq1}\frac{\#\{N\leq X:f(N)\leq\delta N\}}X\to0
 \qquad (\delta\to0).
\]
Thus, for any prescribed $\eta>0$, choosing $\delta$ small enough gives
\[
 \frac{\#\{N\leq X:f(N)>\delta N\}}X\geq 1-\eta
\]
for all $X$ up to the harmless exceptional convention at $2,5$.
\end{proof}

\begin{remark}[Relation to Tao's original $O(\delta^2)$ bound]
Tao's forum argument established the first decisive density obstruction, in the form
\[
 \#\{N\leq X:f(N)\leq \delta N\}\ll \delta^2 X.
\]
One way to view Kova\v{c}'s theorem is that it replaces Tao's first-moment/union-bound estimate by a flexible high-moment estimate.  Liam Price's linked GPT-5.5 Pro write-ups supplied intermediate fixed-power improvements such as $O(\delta^3)$ and $O_k(\delta^k)$, but Kova\v{c}'s note gives a shorter and quantitatively stronger version.
\end{remark}

\section{The claimed limsup resolution}

The thread also contains a June 22, 2026 comment by the user principia\_math, claiming a proposed resolution of the remaining limsup question.

\begin{claim}[principia\_math / Principia Math, as stated on the forum]
For every fixed $A\geq1$, the set of positive integers $N$ satisfying
\[
 f(N)>AN
\]
has positive lower density.  Consequently,
\[
 \limsup_{N\to\infty}\frac{f(N)}N=\infty.
\]
\end{claim}

The forum summary says that the proof was accompanied by an Overleaf LaTeX document and a Lean formalization.  It also says that the Lean file isolates two classical analytic inputs as assumptions: a quantitative Mertens product estimate and an almost-all binary Goldbach theorem.  The same comment notes that the authors later learned of jif's proof of well-definedness and that their Lean formalization instead used an almost-all binary Goldbach estimate.

\begin{remark}
This consolidated note does not reproduce the proof of Claim 4.1.  The reason is not mathematical disagreement, but source availability: the thread text gives only a high-level summary, while the proof itself is in an external Overleaf/Lean link.  Accordingly, Claim 4.1 is recorded as a claimed resolution with its stated dependencies, not as a proof checked in this document.
\end{remark}

\section{What is proved here and what remains source-dependent}

Combining the thread contributions gives the following status.

\begin{enumerate}[label=(\arabic*),leftmargin=2.4em]
\item The values $2$ and $5$ are impossible, and jif's certificate-based proof covers all other positive integers.  Therefore $f(N)$ is well-defined for all $N\notin\{2,5\}$, subject to the finite verifier and explicit analytic inputs cited above.
\item The assertion $f(N)=o(N)$ is false, not merely pointwise but in the almost-all sense: for small $\delta$, the proportion of $N\leq X$ with $f(N)\leq\delta N$ is extremely small, uniformly in $X$.
\item The best density upper bound written out in the thread is Kova\v{c}'s
\[
 \sup_{X\geq1}\frac{\#\{N\leq X:f(N)\leq\delta N\}}X
 \ll \exp\{-\exp((1/\delta)^c)\}.
\]
\item The remaining limsup question is claimed to have been resolved by principia\_math / Principia Math, but its proof is not reproduced in the forum text and therefore is not incorporated as a checked proof here.
\end{enumerate}

\begin{thebibliography}{99}

\bibitem{Bloom1054}
Thomas F. Bloom, \emph{Erd\H{o}s Problem \#1054}, Erd\H{o}s Problems website, problem page and discussion thread, accessed 24 June 2026.\newline
\url{https://www.erdosproblems.com/1054}\newline
\url{https://www.erdosproblems.com/forum/thread/1054}

\bibitem{Guy2004}
Richard K. Guy, \emph{Unsolved Problems in Number Theory}, third edition, Problem B2, Springer, 2004.

\bibitem{TaoForum}
Terence Tao (forum nickname \texttt{TerenceTao}), comment under Erd\H{o}s Problem \#1054, 1 November 2025, giving the density obstruction via the sums $\sigma_j(n)$.

\bibitem{KovacNote}
Vjekoslav Kova\v{c} (forum nickname \texttt{Vjeko\_Kovac}), \emph{An improved bound in Erd\H{o}s Problem \#1054}, short note dated 17 May 2026.\newline
\url{https://web.math.pmf.unizg.hr/~vjekovac/files/Erdos_problem_1054.pdf}

\bibitem{BloomForum}
Thomas Bloom, comments under Erd\H{o}s Problem \#1054, 11 May 2026, on the dependence on the moment parameter and the optimization of the density estimate.

\bibitem{LiamGPT}
Liam Price, comments under Erd\H{o}s Problem \#1054, 10 May 2026, linking GPT-5.5 Pro write-ups with intermediate fixed-power improvements.

\bibitem{jifRepo}
\texttt{jif}, comment under Erd\H{o}s Problem \#1054, 16 April 2026, and GitHub repository \texttt{jif-perso/erdos\_1054}, giving the representability verifier.\newline
\url{https://github.com/jif-perso/erdos_1054}

\bibitem{Helfgott}
Harald A. Helfgott, \emph{The ternary Goldbach conjecture is true}, Annals of Mathematics Studies 195, Princeton University Press, 2015; see especially Section 7 for the explicit estimates used in jif's tail argument.

\bibitem{RosserSchoenfeld}
J. Barkley Rosser and Lowell Schoenfeld, \emph{Approximate formulas for some functions of prime numbers}, Illinois Journal of Mathematics 6 (1962), 64--94.

\bibitem{PrincipiaForum}
\texttt{principia\_math} / Principia Math, comment under Erd\H{o}s Problem \#1054, 22 June 2026, claiming a proposed resolution of the limsup question and linking a LaTeX proof and Lean formalization.

\end{thebibliography}

\end{document}
