Reused writeup on Math-5925 (gpt pro 5.2)

Erdős Problem 830 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/830

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_5.2/830.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/830.tex.

FORMAL RESTATEMENT

For $n\in\mathbb{N}$ let $\sigma(n)=\sum_{d\mid n} d$ be the sum-of-divisors function. A pair $(a,b)\in\mathbb{N}^2$ is called an amicable pair if \[ \sigma(a)=\sigma(b)=a+b. \] Equivalently, writing $s(n)=\sigma(n)-n$ for the sum of proper divisors, amicability is $s(a)=b$ and $s(b)=a$ with $a\ne b$. Questions: 1. Are there infinitely many amicable pairs? 2. If $A(x)$ counts the number of amicable pairs $1\le a\le b\le x$, is it true that $A(x)>x^{1-o(1)}$ as $x\to\infty$?

QUICK LITERATURE/CONTEXT CHECK

The problem statement records that Erdős proved $A(x)=o(x)$ and that Pomerance proved explicit upper bounds of the form $A(x)\le x\exp(- (\log x)^{1/3})$ and later $A(x)\le x\exp(- (\tfrac12+o(1))(\log x\log\log x)^{1/2})$. We do not use these results below.

ATTACK PLAN

Proof track: $1$ Derive structural constraints on amicable pairs from the defining equality $\sigma(a)=a+b$. $2$ Try to construct infinite families by parametrizing $s(n)$ (hard; no complete construction here).

Disproof track: $1$ Look for congruence obstructions forcing finiteness (none found here). $2$ Use computation to gauge growth of $A(x)$ at small $x$.

WORK

Lemma 1 (abundant/deficient dichotomy). Let $(a,b)$ be an amicable pair with $a<b$. Then $a$ is abundant (i.e. $\sigma(a)>2a$) and $b$ is deficient (i.e. $\sigma(b)<2b$).

Proof. Because $\sigma(a)=a+b$ and $b>a$, we have \[ \sigma(a)=a+b> a+a =2a, \] so $a$ is abundant. Similarly, $\sigma(b)=a+b < b+b=2b$ because $a<b$. Thus $b$ is deficient. $\square$

Lemma 2 (no primes in amicable pairs). If $(a,b)$ is an amicable pair, then both $a$ and $b$ are composite integers at least $4$.

Proof. First, $a\ne 1$ because $\sigma(1)=1$ would force $a+b=1$. So $a\ge 2$ and similarly $b\ge 2$.

Assume for contradiction that $a$ is prime, $a=p$. Then $\sigma(p)=1+p$. The amicable condition $\sigma(a)=a+b$ yields $1+p = p+b$, hence $b=1$. But then $\sigma(b)=\sigma(1)=1$, while the defining equality requires $\sigma(b)=a+b=p+1>1$, contradiction. Thus $a$ is not prime. By symmetry, $b$ is not prime. Therefore both are composite. Since the smallest composite integer is $4$, this implies $a,b\ge 4$.

$\square$

Fast reality check (computation; exact search). We computed $A(x)$ for several small values by sieving the sum of proper divisors $s(n)=\sigma(n)-n$ and checking $s(a)=b,\ s(b)=a$. The number of amicable pairs with $1\le a<b\le x$ found was: \[ \begin{array}{r|r} x & A(x) \\\hline 10^4 & 5\\ 5\cdot 10^4 & 8\\ 10^5 & 13\\ 2\cdot 10^5 & 20\\ 5\cdot 10^5 & 28\\ 10^6 & 40 \end{array} \] The first five pairs (for $x\ge 10^4$) were \[(220,284),\ (1184,1210),\ (2620,2924),\ (5020,5564),\ (6232,6368),\] and an odd amicable pair appears by $x=5\cdot 10^4$: \[(12285,14595). \] These data are only sanity checks and do not suggest asymptotics.

VERIFICATION

$1$ Lemma 1 is a direct inequality check and depends only on $a<b$. $2$ Lemma 2: the prime case gives an immediate contradiction; symmetry handles both variables. $3$ The computation uses exact divisor-sum sieving, and the listed amicable pairs match standard small examples.

FINAL

UNRESOLVED

(i) Strongest proved partial result. We proved basic structural facts: in any amicable pair with $a<b$, the smaller number is abundant and the larger is deficient (Lemma 1), and neither entry can be prime (Lemma 2). We also computed $A(x)$ exactly for $x\le 10^6$ (with $A(10^6)=40$).

(ii) First gap (crisp statement). Prove or disprove: there exist infinitely many amicable pairs $(a,b)$.

(iii) Top 3 next moves. 1. Attempt to build parametric families of amicable pairs by controlling $s(n)$ on structured factorizations (e.g. $n=2^u p^v q^w$). 2. Computation: extend exact sieves to larger $x$ (e.g. $10^8$ or more) and fit empirical growth, searching for patterns that could hint at construction. 3. Prove unconditional lower bounds for $A(x)$ beyond triviality; even showing $A(x)\to\infty$ without explicit pairs would be progress.

(iv) Minimal counterexample structure. A counterexample to “infinitely many” would mean there exists $X_0$ such that no amicable pair has both entries $>X_0$. A counterexample to $A(x)>x^{1-o(1)}$ could still allow infinitely many pairs but with very sparse distribution, e.g. $A(x)\le x^{\theta}$ for some fixed $\theta<1$.

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.