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arxiv: 2412.02701 · v4 · submitted 2024-11-10 · 🧮 math.GM

Prime Divisors of 10's Friends: A Generalization of Prior Bounds

Sagar Mandal This is my paper

Pith reviewed 2026-05-23 18:00 UTC · model grok-4.3

classification 🧮 math.GM
keywords abundancy indexfriends of 10solitary numbersprime divisorsupper boundsnumber theorygeneralization
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The pith

Any friend of 10 has every prime divisor bounded above by an explicit function derived from the abundancy equality.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes upper bounds on the prime divisors of any number that shares the abundancy index of 10 but differs from 10 itself. It generalizes an earlier result that had already obtained bounds on the second, third, and fourth smallest such primes. The new work supplies improved upper bounds for the third and fourth smallest prime divisors. A reader would care because these limits restrict the possible forms any friend of 10 could take and thereby narrow the search space in an open question about solitary numbers.

Core claim

If n is a friend of 10, then each prime divisor p of n satisfies an explicit upper bound; moreover, the third and fourth smallest prime divisors of n obey stricter upper bounds than those previously established.

What carries the argument

The condition that the abundancy index of n equals the abundancy index of 10, together with generalized lemmas that bound prime divisors from above.

If this is right

  • Every prime divisor of a friend of 10 is subject to an explicit upper bound.
  • The third smallest prime divisor of any friend of 10 obeys a stricter upper bound than the one given earlier.
  • The fourth smallest prime divisor of any friend of 10 obeys a stricter upper bound than the one given earlier.
  • The bounding technique extends uniformly to all prime divisors rather than stopping at the fourth smallest.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The new bounds reduce the range of integers that must be checked when searching computationally for a friend of 10.
  • The same generalization approach could be applied to obtain bounds on prime divisors of friends of other small solitary candidates.
  • Tighter prime bounds may combine with other known constraints to make an exhaustive search for friends of 10 feasible within a finite range.

Load-bearing premise

Any friend of 10 must satisfy the same abundancy-index equality as 10, and the lemmas imported from the prior paper must hold without gaps.

What would settle it

Existence of an integer n different from 10 with abundancy index exactly equal to that of 10, yet having at least one prime divisor that exceeds one of the derived upper bounds.

read the original abstract

10 is the smallest positive integer which is whether solitary or friendly is still an open question in mathematics. In this paper, we provide upper bounds for each of the prime divisors of a friend of 10. This paper is precisely a generalization of a recent paper [4] in which necessary upper bounds for the 2nd, 3rd, and 4th smallest prime divisors of a friend of 10 have been proved. Further, we establish better upper bounds for the 3rd, and 4th smallest prime divisors of a friend of 10 than the bounds given in [4].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper generalizes results from [4] on numbers n with σ(n)/n = 9/5 (friends of 10). It claims to derive explicit upper bounds on every prime divisor of such an n, together with improved upper bounds on the third- and fourth-smallest prime divisors relative to those already proved in [4]. The derivations rely on case analysis of the prime factorization of n and on lemmas imported from [4] concerning possible forms of the abundancy index.

Significance. If the claimed bounds hold, they tighten the constraints on any hypothetical friend of 10 and could support exhaustive searches or non-existence proofs for such numbers. The extension from bounds on the second-, third-, and fourth-smallest primes to bounds on all primes is a direct and useful generalization of the prior work; the sharpened bounds on the third and fourth primes are presented as concrete improvements.

minor comments (2)
  1. The abstract states that 'better upper bounds' are established for the 3rd and 4th smallest primes but does not record the numerical values of the new bounds or the precise improvement over [4]; adding these values would make the contribution immediately quantifiable.
  2. Notation for the abundancy index σ(n)/n is used without an explicit reminder of the target value 9/5 after the introduction; a single sentence restating the defining equation in §1 would improve readability for readers who have not yet consulted [4].

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No specific major comments were listed in the report, so we have no individual points requiring rebuttal or revision at this stage. We are prepared to address any additional feedback if provided.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript derives explicit upper bounds on prime factors of any n satisfying σ(n)/n = 9/5 by case analysis on the prime factorization form, importing lemmas from the cited prior work [4]. No equation or bound is shown to reduce by construction to a fitted parameter, self-definition, or unverified self-citation chain. The dependence on [4] is ordinary citation of prior lemmas; the new results supply independent extensions (including sharpened bounds on the third- and fourth-smallest primes) that do not collapse to the inputs of [4]. The derivation remains self-contained against the stated abundancy-index assumption and standard multiplicative properties of σ.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5618 in / 1092 out tokens · 45897 ms · 2026-05-23T18:00:23.472794+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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    Chatterjee, S

    T. Chatterjee, S. Mandal and S. Mandal,A note on necessary conditions for a friend of 10. Available online at:https://doi.org/10.48550/arXiv.2404.00624

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    Mandal and S

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    work page 2025

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    work page 2025

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    Available online at:https://oeis.org/A074902

    OEIS Foundation Inc.,The Online Encyclopedia of Integer Sequences, Sequence A074902, Ac- cessed October 2024. Available online at:https://oeis.org/A074902

    work page 2024

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    work page 2000