Reused writeup on Math-6114 (gpt pro 5.2)

Erdős Problem 1060 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/1060

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/1060.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/1060.tex.

FORMAL RESTATEMENT

Let $\sigma(k)=\sum_{d\mid k} d$ be the sum of divisors of $k\ge 1$. For $n\ge 1$ define \[ f(n):=\#\{k\in\mathbb Z_{\ge 1}:\ k\sigma(k)=n\}. \] The problem asks whether \[ f(n)\le n^{o(1/\log\log n)}\quad\text{as }n\to\infty, \] and perhaps even $f(n)\le (\log n)^{O(1)}$.

QUICK LITERATURE/CONTEXT CHECK

The problem is posed as an open question (Guy B11). I do not use or claim any results not stated in the problem text.

ATTACK PLAN

Proof-track: derive structural constraints on $k$ from $k\sigma(k)=n$ (e.g. $k\mid n$, size bounds, multiplicativity) and try to show the equation has few solutions. Disproof-track: search for $n$ with unusually many representations $n=k\sigma(k)$ by enumerating $k$ and counting collisions.

WORK

FAST REALITY CHECK. For $n\le 10{,}000$, a brute-force check over $k\le \lfloor\sqrt n\rfloor$ finds:

B=10000 max f(n)=2
n=336 has solutions k=[12,14]
n=5952 has solutions k=[48,62]

Enumerating all $k\le 10^6$ and counting values of $k\sigma(k)$ gives the following empirical maximum multiplicity:

max f(n) observed for k<=1,000,000: 5
achieved at n=1584858562560
with solutions k=[624960, 640080, 696384, 708660, 713232]

Lemma 1060.1 (divisor and size constraints). If $n>1$ and $k\sigma(k)=n$, then (a) $k\mid n$, and (b) $k<\sqrt n$.

Proof. (a) The equation $k\sigma(k)=n$ implies $k$ divides the left-hand side, hence $k\mid n$. (b) For every $k\ge 1$ we have $\sigma(k)\ge k$ because $k$ itself is a divisor of $k$; if $k>1$ then $\sigma(k)\ge 1+k>k$. Thus for $n>1$ any solution has $k\ge 2$ and \[ n=k\sigma(k)\ge k\cdot k = k^2, \] with strict inequality $n>k^2$ since $\sigma(k)>k$ for $k>1$. Hence $k<\sqrt n$. $\square$

Lemma 1060.2 (trivial divisor-function upper bound). For $n>1$, \[ f(n)\le \frac{d(n)}{2}, \] where $d(n)$ is the number of positive divisors of $n$.

Proof. By Lemma 1060.1(a), every solution $k$ is a divisor of $n$. By Lemma 1060.1(b), every solution satisfies $k<\sqrt n$. Among all divisors of $n$, the number of divisors strictly less than $\sqrt n$ is at most $d(n)/2$ because divisors pair as $k\leftrightarrow n/k$, with exactly one divisor at $\sqrt n$ only when $n$ is a perfect square. Therefore $f(n)$, which counts only divisors $k<\sqrt n$ satisfying the equation, is at most $d(n)/2$. $\square$

VERIFICATION

Lemma 1060.1 uses only the basic fact that $k$ divides $k\sigma(k)$ and that $\sigma(k)\ge k$ with strict inequality for $k>1$. Lemma 1060.2 checks the only edge case where $k=\sqrt n$ could occur (perfect squares): Lemma 1060.1(b) rules out $k=\sqrt n$ for $n>1$ because $n>k^2$ for $k>1$. The brute-force computations explicitly restricted to $k\le \sqrt n$ (for $n\le 10^4$) and to $k\le 10^6$ (for the larger collision search), matching Lemma 1060.1(b).

FINAL

UNRESOLVED

1. (i) Strongest proved partial result. Every solution $k$ of $k\sigma(k)=n$ satisfies $k\mid n$ and $k<\sqrt n$, hence $f(n)\le d(n)/2$ for $n>1$ (Lemmas 1060.1--1060.2). Empirically, for $k\le 10^6$ the largest multiplicity observed is $f(n)=5$ at $n=1584858562560$.
2. (ii) First gap (crisp). Prove a nontrivial asymptotic upper bound of the form $f(n)\le (\log n)^{C}$ for all sufficiently large $n$, or even $f(n)\le \exp(o(\log n/\log\log n))$.
3. (iii) Top 3 next moves. $1$ Structural: classify solutions when $k$ has restricted prime factorization (e.g. prime powers), and show that each such class contributes very few $k$ to a fixed $n$. $2$ Collision control: prove that the map $k\mapsto k\sigma(k)$ is "almost injective" on large ranges by showing that $k\sigma(k)$ has a rigid prime-factor pattern determined by $k$. $3$ Computation: search for $n$ with larger $f(n)$ by enumerating $k$ beyond $10^6$ using a fast divisor-sum sieve and tracking multiplicities.
4. (iv) Minimal counterexample structure. A counterexample to polylogarithmic bounds would require some $n$ having many distinct divisors $k<\sqrt n$ with $\sigma(k)=n/k$. Such an $n$ would plausibly be very "smooth" (many small prime factors) to admit many candidate divisors $k$.

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.