Reused writeup on Math-970 (gpt pro 5.2)

Erdős Problem 886 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/886

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_5.2/886.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/886.tex.

Let $\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$? Erdős attributes this conjecture to Ruzsa. Erdős and Rosenfeld [ErRo97] proved that there are infinitely many $n$ such that there are four divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/4})$.

1) FORMAL RESTATEMENT

Fix $\epsilon>0$. For each integer $n\ge 1$, let \[I_{\epsilon}(n):=\big(\sqrt n,\ \sqrt n+n^{1/2-\epsilon}\big). \] Let \[r_{\epsilon}(n):=\big|\{d\mid n:\ d\in I_{\epsilon}(n)\}\big|. \] Question: Is it true that for each $\epsilon>0$ there exist constants $C(\epsilon)$ and $n_0(\epsilon)$ such that \[r_{\epsilon}(n)\le C(\epsilon)\quad\text{for all integers }n\ge n_0(\epsilon)? \]

2) QUICK LITERATURE/CONTEXT CHECK

The problem statement notes: - Erdős attributes the conjecture to Ruzsa. - Erdős--Rosenfeld $1997$ proved there are infinitely many $n$ with $r_{1/4}(n)\ge 4$ (i.e. four divisors in $(\sqrt n,\sqrt n+n^{1/4})$). I do not use any additional external results.

3) ATTACK PLAN

- Relate divisors in $(\sqrt n,\sqrt n+H)$ to “near-square” factorizations $n=ab$ with $b-a$ small. - Prove deterministic inequalities translating divisor-location bounds into bounds on factor differences. - Compute $r_{\epsilon}(n)$ for $n$ up to a moderate range to sanity-check growth.

4) WORK

Lemma 886.1 (Divisors just above $\sqrt n$ correspond to near-equal factor pairs). Let $n\ge 1$ and let $H$ satisfy $0<H\le \sqrt n$. If $d\mid n$ and \[\sqrt n<d<\sqrt n+H,\] then with $a:=n/d$ (an integer) one has $a<\sqrt n$ and \[0<d-a<2H+\frac{H^2}{\sqrt n} \le 3H. \] In particular, every such divisor gives a factorization $n=ab$ with $a<b$ and $b-a<3H$.

Proof. Let $d\mid n$ with $d>\sqrt n$ and set $a=n/d\in\mathbb N$. Then $a<\sqrt n$ since $ad=n<d^2$. Also \[d-a = d-\frac{n}{d}=\frac{d^2-n}{d}. \] Because $d<\sqrt n+H$, \[d^2-n < (\sqrt n+H)^2-n = 2H\sqrt n+H^2. \] Dividing by $d>\sqrt n$ gives \[d-a=\frac{d^2-n}{d} < \frac{2H\sqrt n+H^2}{\sqrt n}=2H+\frac{H^2}{\sqrt n}. \] If additionally $H\le \sqrt n$, then $H^2/\sqrt n\le H$, hence $d-a<3H$. $\square$

Lemma 886.2 (Localization of the complementary divisor). Under the assumptions of Lemma 886.1 with $H\le \sqrt n$, the complementary divisor $a=n/d$ satisfies \[\sqrt n-3H<a<\sqrt n. \]

Proof. From Lemma 886.1, $d-a<3H$, i.e. $a>d-3H$. Since $d>\sqrt n$, this gives $a>\sqrt n-3H$. Also $a<\sqrt n$ was proved in Lemma 886.1. $\square$

Lemma 886.3 (Connection to the factor-difference set $D(n)$). Let $n\ge 1$ and $H\le \sqrt n$. If $d\mid n$ lies in $(\sqrt n,\sqrt n+H)$ and $a=n/d$, then the factor difference $\delta:=d-a$ satisfies \[\delta\in D(n)\quad\text{and}\quad 0<\delta<3H. \] Consequently, \[r_{\epsilon}(n)\le \big|D(n)\cap\{1,2,\dots,\lfloor 3n^{1/2-\epsilon}\rfloor\}\big|. \]

Proof. The factorization $n=ad$ shows $|d-a|\in D(n)$ by definition of $D(n)$ (Problem #885). Lemma 886.1 gives $0<d-a<3H$. Taking $H=n^{1/2-\epsilon}$ yields the displayed bound on $r_{\epsilon}(n)$. $\square$

FAST REALITY CHECK (computation). For $n\le 10^6$, the maximum observed values of $r_{\epsilon}(n)$ for several $\epsilon$ were: - $\epsilon=0.25$: max $r_{0.25}(n)=1$ (no instance with $\ge 2$ divisors in $(\sqrt n,\sqrt n+n^{1/4})$ up to $10^6$). - $\epsilon=0.20$: max $r_{0.20}(n)=4$ at $n=712800$, with divisors in the interval equal to $\{864,880,891,900\}$. - $\epsilon=0.15$: max $r_{0.15}(n)=5$ at $n=194040$, with divisors $\{441,462,490,495,504\}$. - $\epsilon=0.10$: max $r_{0.10}(n)=9$ at $n=720720$, with divisors $\{858,880,910,924,936,990,1001,1008,1040\}$. These computations are only sanity checks and do not address the asymptotic question.

5) VERIFICATION

- Lemma 886.1: verified the inequality chain carefully; the only place a constant appears is when using $H\le\sqrt n$ to deduce $H^2/\sqrt n\le H$. - Lemma 886.3: checked that the mapping from a divisor $d>\sqrt n$ to the factor difference $d-n/d$ is well-defined and produces an element of $D(n)$. - Computation: ensured the interval is open; since divisors are integers, boundary inclusion does not affect counts unless $\sqrt n$ is integer or $\sqrt n+H$ integer, but those only change the count by at most 1. The reported divisors are strictly between the endpoints numerically.

6) FINAL

UNRESOLVED

(i) Strongest proved partial result. - Divisors in $(\sqrt n,\sqrt n+H)$ correspond to factor pairs $n=ab$ with $b-a<3H$ (Lemmas 886.1--886.2). - Such divisors inject into small values of the factor-difference set $D(n)$ (Lemma 886.3), linking this problem to #885. - Explicit computations up to $10^6$ give small maxima for several $\epsilon$ values.

(ii) First gap (crisp). For fixed $\epsilon>0$, prove or disprove the existence of an absolute bound $C(\epsilon)$ for \[r_{\epsilon}(n)=|\{d\mid n:\ \sqrt n<d<\sqrt n+n^{1/2-\epsilon}\}|\] that holds for all sufficiently large $n$.

(iii) Top 3 next moves. 1. Use Lemma 886.3 to attempt to bound the number of small values in $D(n)$ uniformly in $n$ for a given scaling $n^{1/2-\epsilon}$. 2. Try to construct families of $n$ with many divisors near $\sqrt n$ by forcing many factorizations $n=a(a+\delta)$ with $\delta\ll n^{1/2-\epsilon}$; quantify how many distinct $\delta$ can coexist. 3. Extend computation to search for instances with $r_{1/4}(n)\ge 4$ (known to exist infinitely often per the statement) to guess structural patterns.

(iv) Minimal counterexample structure. If the conjecture is false for some $\epsilon$, a minimal counterexample would be a sequence $n_m\to\infty$ with $r_{\epsilon}(n_m)\to\infty$. By Lemma 886.1 this would yield infinitely many distinct factorizations $n_m=a(a+\delta)$ with $\delta\ll n_m^{1/2-\epsilon}$, i.e. $n_m$ admitting many “near-square” factorizations with very small factor gaps.

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.