PaperlayerView exact revision on arXiv

Source-linked AI summary

Amicable numbers

Paul Pollack

arXiv:2610.05661v1math.NT

TL;DR

The paper addresses the distribution of amicable numbers, for which infinitude is unproved despite many known pairs. It establishes a new upper bound using a proof discovered by ChatGPT Astra, and presents it as the first bound of the correct shape.

  • Problem

    Infinitude of amicable numbers remains unproved, despite more than a billion known amicable pairs.

    Source: §1 · p. 1

  • Method

    The proof fixes one pair member and a factor of the other with a large common divisor in their divisor sums, then counts possible choices.

    Source: §2 · p. 6

  • Results

    The paper establishes a new upper bound for A(x), described as the first upper bound of the correct shape.

    Source: §1 · pp. 1–2

  • Takeaways & Limitations

    A heuristic based on multiplicities of σ-values suggests the correct shape and that the constant cannot be replaced by anything larger than 2.

    Source: §1 · p. 2

    One proof case assumes condition (2), bounding the smooth-part contribution and the rough-part prime-factor count by (1/2 − 4ε) log x.

    Source: §2 · p. 8

Abstract

from arXiv · show

Let $s(n):=\sum_{d\mid n,\,d < n} d$ denote the sum of the proper divisors of $n$. Distinct positive integers $n,m$ form an amicable pair if $s(n)=m$ and $s(m)=n$. Let $A(x)$ count the positive integers not exceeding $x$ that belong to an amicable pair. We report on a proof, found by ChatGPT Astra, that $$ A(x)\le x\exp\{-(1/2+o(1))\log x\,\log_3 x/\log_2 x\}$$ as $x\to\infty$. Here $\log_k$ denotes the $k$th iterate of the natural logarithm.

1. Introduction

The introduction defines amicable numbers and frames their distribution as an open problem: infinitude is unproved despite many known pairs. It presents a new upper bound, expected to have the correct shape, and explains the heuristic behind that expectation.

1. Introduction

  • Infinitude of amicable numbers remains unproved, although over a billion amicable pairs are known.A(x) counts amicable numbers not exceeding x.Source: §1 · p. 1
  • The authors establish a new upper bound for A(x), whose proof was discovered by ChatGPT Astra.They describe it as the first upper bound of the “correct shape.”Source: §1 · pp. 1–2
  • A heuristic using large multiplicities of σ-values suggests about x/L(x)^{2+o(1)} amicable pairs and that the constant 2 cannot be exceeded.The heuristic draws on solutions to ϕ(m)=n and σ(m)=n, and on pairs sharing a common divisor sum.Source: §1 · p. 2
  • A second-moment bound for Nσ suggests A(x) ≤ x/L(x)^{2+o(1)} if the condition n+n′=m holds for about an x^-1 proportion of pairs.The introduction derives the conditional estimate from a bound on pairs with equal σ-values.Source: §1 · p. 2

Definitions and notation.

  • The notation distinguishes the y-smooth and y-rough parts of n, and defines a unitary divisor d by d | n and (d, n/d) = 1.The paper writes these parts as n≤y and n>y, respectively.Source: Definitions and notation · p. 2

Statement of AI usage.

  • P.P. says they edited Astra’s proof explanation and take responsibility for the mathematical content without claiming discovery credit.Astra also produced a Lean formalization whose proof was checked by Lean’s kernel.Source: Statement of AI usage · pp. 2–3

2. Proof of Theorem 1

The proof reduces most amicable pairs to a structured factorization and counts them using shared divisors of divisor sums, with the final counting argument split according to the largest prime factor of a squarefree component. It concludes by bounding admissible solutions to obtain the stated bound for A(x).

Auxiliary lemmas

  • Lemma 2 bounds the number of squarefree b ≤ Y for which log(σ(b)≤y) + (log_2 x)Ω(σ(b)>y) exceeds κ log x by YL(x)^(-κ+o(1)).It holds uniformly for 1 ≤ Y ≤ X, with X = x log x and y = log x.Source: §2 · p. 3

Proposition 3

  • Apart from at most xL(x)^(-1/2+o(1)) exceptions, each relevant amicable pair admits the factorization in Proposition 3, with P+(a′) < P+(a) ≤ x^(4/5).The proposition reduces the proof to pairs with large, relatively prime squarefree divisors whose large prime factors are controlled.Source: §2 · p. 4

Proof of Theorem 1

  • The proof roadmap fixes one pair member as mR and a factor v of the other, then uses a large common divisor of σ(m) and σ(v) to restrict choices.After fixing m and v, the amicable relations allow at most two admissible values of R.Source: §2 · p. 6
  • In Case I, P+(a) > x^(ε²), the counting argument saves a power of x; Case II treats P+(a) ≤ x^(ε²).For Case II, the proof factors a′ into bounded-size pieces and averages to choose V and d with d dividing both σ(V) and σ(b), then uses a congruence to restrict R.Source: §2 · pp. 7–9
  • Any two admissible solutions R are equal, since their difference yields a constraint incompatible with V > x^(2ε+3ε²).The passage states that this uniqueness follows as in Case I.Source: §2 · p. 9
  • For fixed d, the proof estimates the sum of 1/V over squarefree V ≤ X satisfying d | σ(V), using a variant of a Brun–Titchmarsh argument.The cited variants are [11, Lemma 2.4], [9, Lemma 2.1], and [12, Lemma 7].Source: §2 · p. 9
  • For fixed m and d, at most x^(ε−4ε³+o(1)) corresponding V occur, and each m,V determines at most one admissible R and n.Summing over m ≤ Xx^(-ε) and d | σ(m), with at most τ(σ(m)) = x^o(1) choices of d, bounds the nonexceptional n in Case II.Source: §2 · p. 10
  • For every fixed 0 < ε < 1/10, A(x) ≤ xL(x)^(-1/2+4ε+o(1)), and the theorem follows.Each amicable pair contributes at most two amicable numbers to A(x).Source: §2 · p. 11
Loading 2610.05661v1…