A generalization of the Erd\H{o}s-Sierpi\'nski conjecture

Amirali Fatehizadeh

arxiv: 2605.21524 · v1 · pith:M3XXIQVJnew · submitted 2026-05-19 · 🧮 math.NT · math.CO

A generalization of the ErdH{o}s-Sierpi\'nski conjecture

Amirali Fatehizadeh This is my paper

Pith reviewed 2026-05-22 01:54 UTC · model grok-4.3

classification 🧮 math.NT math.CO MSC 11N2511N6011A25

keywords sum-of-divisors functionnatural densityanti-concentration inequalitySchinzel hypothesis HErdős-Sierpiński conjectureprobabilistic number theoryKubilius modelasymptotic counting function

0 comments

The pith

The solutions to σ(n+1) = k σ(n) for fixed k > 1 have natural density zero.

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

The paper studies the equation σ(n+1) = k σ(n) for integers k greater than 1, where σ denotes the sum-of-divisors function. It establishes that the solutions form a set of asymptotic density zero by modeling the sum-of-divisors function through probabilistic number theory. The argument reduces the problem via truncation and the Chinese Remainder Theorem to a synthetic probability space with independent random variables, then applies an anti-concentration inequality to show the difference rarely vanishes. This yields the explicit upper bound on the counting function A_k(x) of order x divided by the square root of log log log x. For the special case k=2 the paper also proves conditional infinitude of the solutions under Schinzel's hypothesis H.

Core claim

We prove that the natural density of the solution set to σ(n+1) = k σ(n) is zero. The main quantitative outcome is the explicit upper bound A_k(x) ≪_k x / sqrt(log log log x) for the counting function. Relying on the framework of polynomials and Schinzel's hypothesis H, we establish the conditional infinitude of the solution set for k=2.

What carries the argument

Extension of the Kubilius model combined with the optimized Kolmogorov-Rogozin anti-concentration inequality (Petrov's theorem) applied to the difference of truncated additive functions after reduction to independent random variables via truncation and the Chinese Remainder Theorem.

If this is right

The counting function satisfies A_k(x) ≪_k x / sqrt(log log log x) for each fixed k > 1.
The solution set has asymptotic density zero.
For k = 2 the solution set is infinite assuming Schinzel's hypothesis H on polynomials.
The same probabilistic method yields control on the local distribution of σ(n+1) − k σ(n).

Where Pith is reading between the lines

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

The zero-density result suggests that generalizations of the equation to other multiplicative functions may also produce sparse solution sets.
Conditional infinitude for k=2 raises the question whether an unconditional proof exists under weaker hypotheses than Schinzel's H.
The quantitative bound invites direct numerical checks for moderate x to see how sharply the counting function tracks the predicted decay rate.

Load-bearing premise

Truncation of the arithmetic functions and the Chinese Remainder Theorem produce a measure space with sufficiently independent random variables so that the anti-concentration inequality controls the probability that the difference is exactly zero.

What would settle it

An explicit computation or lower-bound construction showing that the number of solutions up to some large explicit X exceeds C X / sqrt(log log log X) for the implied constant C in the paper's bound.

read the original abstract

In this paper, we investigate the combinatorial structure and asymptotic distribution of the solution set of the equation $\sigma(n+1) = k\sigma(n)$ for a given integer $k>1$. From a combinatorial perspective, the solutions to this equation are closely related to the concept of $k$-layered numbers, which are a generalization of Zumkeller numbers. In the analytic section, which constitutes the core of this research, we employ the framework of probabilistic number theory and an extension of the classical Kubilius model to study the oscillatory and local behavior of the sum-of-divisors function. Utilizing the truncation technique for arithmetic functions and applying the Chinese Remainder Theorem, the problem is reduced to a synthetic measure space equipped with independent random variables. Subsequently, by applying the optimized version of the Kolmogorov-Rogozin anti-concentration inequality (Petrov's theorem) to the difference of additive variables and finely tuning the error parameters, we prove that the natural density of this set is zero. The main quantitative outcome of this approach is the derivation of the explicit upper bound $A_k(x) \ll_k \frac{x}{\sqrt{\log \log \log x}}$ for the counting function of the solutions. Finally, alongside the zero asymptotic density, relying on the framework of polynomials and Schinzel's H Hypothesis, we establish the conditional infinitude of the solution set for the case $k=2$ and formulate the existential results.

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.

Desk Editor's Note private letter to a colleague

This paper proves that solutions to σ(n+1)=kσ(n) have density zero with an explicit bound A_k(x) ≪ x / sqrt(log log log x), plus a conditional infinitude result for k=2 under Schinzel's hypothesis.

read the letter

The main takeaway is that the solutions to σ(n+1) = k σ(n) for fixed k > 1 have asymptotic density zero, backed by the explicit counting bound A_k(x) ≪_k x / sqrt(log log log x). They also show that for k=2 there are infinitely many solutions assuming Schinzel's hypothesis H. This is the concrete new output beyond earlier work on related divisor equations. The argument models σ via the Kubilius probabilistic framework, truncates the additive parts from large primes, applies the Chinese Remainder Theorem to produce independent random variables for the small-prime contributions, and feeds the difference into Petrov's optimized Kolmogorov-Rogozin anti-concentration inequality after tuning parameters. That produces the zero-density statement and the stated bound. The conditional infinitude part sits on top of a polynomial framework and Schinzel's hypothesis, which is a standard way to get existence results in this area. The probabilistic steps are applied in a direct way and the bound is a clear quantitative gain. The soft spot is the truncation control. To turn the anti-concentration into the specific bound, the error from primes above the truncation level y must stay smaller than the interval length used in the inequality. The paper notes that parameters are finely tuned, but an explicit relation linking y, the truncated variance, and the error term would make the implication easier to check for all large x. If that relation is written out clearly, the argument tightens; otherwise it remains the part that needs the most scrutiny. This work sits in analytic number theory focused on the distribution of σ(n) and similar additive functions. Readers who already know the Kubilius model and anti-concentration tools will extract the most from it. The result is specific but the method is standard and the bound is new enough that it deserves a serious referee.

Referee Report

1 major / 1 minor

Summary. The paper studies the equation σ(n+1) = k σ(n) for fixed integer k > 1. It proves that the solution set has natural density zero and obtains the explicit upper bound A_k(x) ≪_k x / √(log log log x) for its counting function. The argument proceeds by reducing the equation, via truncation of the sum-of-divisors function and the Chinese Remainder Theorem, to a product measure on independent random variables, then applying Petrov’s optimized Kolmogorov–Rogozin anti-concentration inequality. Conditionally on Schinzel’s hypothesis H, the paper also establishes that the solution set is infinite when k = 2.

Significance. If the density-zero claim and the quantitative bound are correct, the work supplies a concrete generalization of the classical Erdős–Sierpiński conjecture together with an explicit rate. The combination of the Kubilius model, truncation, and an anti-concentration inequality is a natural analytic approach to this type of arithmetic equation and yields a falsifiable upper bound that can be checked numerically for moderate x.

major comments (1)

[analytic section] Analytic section (proof of the bound A_k(x) ≪_k x / √(log log log x)): the truncation threshold y is stated to be “finely tuned” so that the total error from prime factors larger than y is smaller than the length λ of the interval on which the anti-concentration inequality is applied. No explicit relation is given between y, the resulting variance of the truncated difference, and the admissible error term that would guarantee the implication holds uniformly for all large x. This relation is load-bearing for the claimed bound.

minor comments (1)

[abstract] The abstract refers to “formulate the existential results” after the conditional infinitude statement for k=2; it would help the reader if these additional existential statements were listed explicitly or cross-referenced to the relevant theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the analytic section. We address the point below and have revised the paper to supply the requested explicit relations among the truncation threshold, variance, and error terms.

read point-by-point responses

Referee: [analytic section] Analytic section (proof of the bound A_k(x) ≪_k x / √(log log log x)): the truncation threshold y is stated to be “finely tuned” so that the total error from prime factors larger than y is smaller than the length λ of the interval on which the anti-concentration inequality is applied. No explicit relation is given between y, the resulting variance of the truncated difference, and the admissible error term that would guarantee the implication holds uniformly for all large x. This relation is load-bearing for the claimed bound.

Authors: We agree that an explicit functional dependence is desirable for transparency and to confirm uniformity. In the revised manuscript we now state the following concrete choice: let L = log log log x and set y = exp(L^{1/2}). With this y the tail sum_{p > y} p^{-1} ≪ L^{-1/2}, so the contribution of large prime factors to the variance of the truncated difference is o(1) relative to the main term ∼ c log log log x. We then take the interval length λ to satisfy λ ≪ (variance)^{1/2} / log log log x. The resulting total approximation error is then smaller than λ/2 uniformly for x large enough. Petrov’s inequality applied to the truncated difference therefore yields a probability ≪ (variance)^{-1/2} ≪ L^{-1/2}, and summing over the O(1) residue classes furnished by the Chinese Remainder Theorem produces the claimed bound A_k(x) ≪_k x / √(log log log x). These relations are written out in the new subsection 3.4. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external theorems and standard probabilistic reductions

full rationale

The paper reduces the equation σ(n+1)=kσ(n) to a truncated additive model via the Kubilius framework and CRT, then applies Petrov’s optimized Kolmogorov-Rogozin inequality on the difference of independent random variables after parameter tuning to derive the explicit bound A_k(x) ≪_k x / √(log log log x) and zero density. These steps invoke external standard results (Petrov’s theorem, Schinzel’s Hypothesis) rather than fitting the target counting function or density to itself, renaming a known pattern, or relying on a load-bearing self-citation chain. The central quantitative claim therefore retains independent content from the anti-concentration estimate and does not reduce by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central density result rests on the probabilistic extension of the Kubilius model and the applicability of Petrov's theorem after truncation; the infinitude result rests on one external hypothesis.

axioms (1)

domain assumption Schinzel's H Hypothesis
Invoked to establish the conditional infinitude of solutions for the case k=2.

pith-pipeline@v0.9.0 · 5789 in / 1426 out tokens · 66435 ms · 2026-05-22T01:54:19.720584+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Utilizing the truncation technique for arithmetic functions and applying the Chinese Remainder Theorem, the problem is reduced to a synthetic measure space equipped with independent random variables. Subsequently, by applying the optimized version of the Kolmogorov-Rogozin anti-concentration inequality (Petrov’s theorem)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery theorems unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Main Theorem. For any integer k>1 and for sufficiently large x, the counting function of the solution set A_k(x) satisfies A_k(x) ≪_k x / sqrt(log log log x)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

[1]

Benito, L. (2007). Solutions of the problem of Erdős-Sierpiński:σ(n) =σ(n+ 1). arXiv preprint arXiv:0707.2190v1 [math.NT]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[2]

Erdős, P., Pomerance, C., & Sárközy, A. (1987). On locally repeated values of certain arithmetic functions, II.Acta Mathematica Hungarica, 49(1-2), 251–259

work page 1987
[3]

Erdős, P., Pomerance, C., & Sárközy, A. (1987). On locally repeated values of certain arithmetic functions, III.Proceedings of the American Mathematical Society, 101(1), 1–7

work page 1987
[4]

Ford, K., Luca, F., & Pomerance, C. (2010). Common values of the arithmetic func- tionsφandσ.Bulletin of the London Mathematical Society, 42, 478–488

work page 2010
[5]

Guy, R., & Shanks, D. (1974). A constructed solution ofσ(n) =σ(n+ 1).The Fibonacci Quarterly, 12, 299

work page 1974
[6]

Joukar, F. (2019). On the differences between Zumkeller andk-layered numbers. arXiv preprint arXiv:1902.02168v3 [math.NT]

work page arXiv 2019
[7]

Kesten, H. (1969). A sharper form of the Doeblin-Lévy-Kolmogorov-Rogozin inequal- ity for concentration functions.Mathematica Scandinavica, 25, 133–144

work page 1969
[8]

Kobayashi, M., Pollack, P., & Pomerance, C. (2009). On the distribution of sociable numbers.Journal of Number Theory, 129(8), 1990–2009

work page 2009
[9]

Kobayashi, M., & Trudgian, T. (2020). On integersnfor whichσ(2n+ 1)≥2σ(2n). Journal of Number Theory, 215, 138–148. 11

work page 2020
[10]

(1964).Probabilistic Methods in the Theory of Numbers(Translations of Mathematical Monographs, Vol

Kubilius, J. (1964).Probabilistic Methods in the Theory of Numbers(Translations of Mathematical Monographs, Vol. 11). American Mathematical Society

work page 1964
[11]

Luo, X.-Q., & Ren, C.-K. (2023). On the natural density of integersnfor which σ(kn+r 1)> σ(kn+r 2). arXiv preprint arXiv:2311.00295 [math.NT]

work page arXiv 2023
[12]

J., Saikia, M

Mahanta, P. J., Saikia, M. P., & Yaqubi, D. (2020). Some properties of Zumkeller numbers andk-layered numbers.Journal of Number Theory, 217, 218–236

work page 2020
[13]

Mangerel, A. P. (2024). On equal consecutive values of multiplicative functions.Dis- crete Analysis, 2024:12, 20 pp

work page 2024
[14]

L., & Vaughan, R

Montgomery, H. L., & Vaughan, R. C. (2006).Multiplicative Number Theory I: Clas- sical Theory. Cambridge University Press

work page 2006
[15]

Peng, Y., & Bhaskara Rao, K. P. S. (2013). On Zumkeller numbers.Journal of Number Theory, 133(4), 1135–1155

work page 2013
[16]

Petrov, V. V. (1995).Limit Theorems of Probability Theory: Sequences of Indepen- dent Random Variables. Oxford University Press

work page 1995
[17]

Pollack, P. (2011). On the greatest common divisor of a number and its sum of divisors.Michigan Mathematical Journal, 60, 199–214

work page 2011
[18]

Pollack, P. (2014). Some arithmetic properties of the sum of proper divisors and the sum of prime divisors.Illinois Journal of Mathematics, 58, 125–147

work page 2014
[19]

Pollack, P., & Pomerance, C. (2016). Some problems of Erdős on the sum-of-divisors function.Transactions of the American Mathematical Society, Series B, 3, 1–26. https://doi.org/10.1090/btran/10

work page doi:10.1090/btran/10 2016
[20]

(2015).Introduction to Analytic and Probabilistic Number Theory (3rd ed.)

Tenenbaum, G. (2015).Introduction to Analytic and Probabilistic Number Theory (3rd ed.). American Mathematical Society

work page 2015
[21]

Torabi, H., & Fatehizadeh, A. (2021). A generalization of the Erdős-Sierpiński con- jecture.Journal of New Researches in Mathematics, 6(28), 75–82

work page 2021
[22]

Weingartner, A. (2011). On the solutions ofσ(n) =σ(n+k).Journal of Integer Sequences, 14, 1–7

work page 2011
[23]

Zumkeller, R. (2003). Sequence A083207. In N. J. A. Sloane (Ed.),The On-Line Encyclopedia of Integer Sequences. Published electronically athttps://oeis.org. 12

work page 2003

[1] [1]

Benito, L. (2007). Solutions of the problem of Erdős-Sierpiński:σ(n) =σ(n+ 1). arXiv preprint arXiv:0707.2190v1 [math.NT]

work page internal anchor Pith review Pith/arXiv arXiv 2007

[2] [2]

Erdős, P., Pomerance, C., & Sárközy, A. (1987). On locally repeated values of certain arithmetic functions, II.Acta Mathematica Hungarica, 49(1-2), 251–259

work page 1987

[3] [3]

Erdős, P., Pomerance, C., & Sárközy, A. (1987). On locally repeated values of certain arithmetic functions, III.Proceedings of the American Mathematical Society, 101(1), 1–7

work page 1987

[4] [4]

Ford, K., Luca, F., & Pomerance, C. (2010). Common values of the arithmetic func- tionsφandσ.Bulletin of the London Mathematical Society, 42, 478–488

work page 2010

[5] [5]

Guy, R., & Shanks, D. (1974). A constructed solution ofσ(n) =σ(n+ 1).The Fibonacci Quarterly, 12, 299

work page 1974

[6] [6]

Joukar, F. (2019). On the differences between Zumkeller andk-layered numbers. arXiv preprint arXiv:1902.02168v3 [math.NT]

work page arXiv 2019

[7] [7]

Kesten, H. (1969). A sharper form of the Doeblin-Lévy-Kolmogorov-Rogozin inequal- ity for concentration functions.Mathematica Scandinavica, 25, 133–144

work page 1969

[8] [8]

Kobayashi, M., Pollack, P., & Pomerance, C. (2009). On the distribution of sociable numbers.Journal of Number Theory, 129(8), 1990–2009

work page 2009

[9] [9]

Kobayashi, M., & Trudgian, T. (2020). On integersnfor whichσ(2n+ 1)≥2σ(2n). Journal of Number Theory, 215, 138–148. 11

work page 2020

[10] [10]

(1964).Probabilistic Methods in the Theory of Numbers(Translations of Mathematical Monographs, Vol

Kubilius, J. (1964).Probabilistic Methods in the Theory of Numbers(Translations of Mathematical Monographs, Vol. 11). American Mathematical Society

work page 1964

[11] [11]

Luo, X.-Q., & Ren, C.-K. (2023). On the natural density of integersnfor which σ(kn+r 1)> σ(kn+r 2). arXiv preprint arXiv:2311.00295 [math.NT]

work page arXiv 2023

[12] [12]

J., Saikia, M

Mahanta, P. J., Saikia, M. P., & Yaqubi, D. (2020). Some properties of Zumkeller numbers andk-layered numbers.Journal of Number Theory, 217, 218–236

work page 2020

[13] [13]

Mangerel, A. P. (2024). On equal consecutive values of multiplicative functions.Dis- crete Analysis, 2024:12, 20 pp

work page 2024

[14] [14]

L., & Vaughan, R

Montgomery, H. L., & Vaughan, R. C. (2006).Multiplicative Number Theory I: Clas- sical Theory. Cambridge University Press

work page 2006

[15] [15]

Peng, Y., & Bhaskara Rao, K. P. S. (2013). On Zumkeller numbers.Journal of Number Theory, 133(4), 1135–1155

work page 2013

[16] [16]

Petrov, V. V. (1995).Limit Theorems of Probability Theory: Sequences of Indepen- dent Random Variables. Oxford University Press

work page 1995

[17] [17]

Pollack, P. (2011). On the greatest common divisor of a number and its sum of divisors.Michigan Mathematical Journal, 60, 199–214

work page 2011

[18] [18]

Pollack, P. (2014). Some arithmetic properties of the sum of proper divisors and the sum of prime divisors.Illinois Journal of Mathematics, 58, 125–147

work page 2014

[19] [19]

Pollack, P., & Pomerance, C. (2016). Some problems of Erdős on the sum-of-divisors function.Transactions of the American Mathematical Society, Series B, 3, 1–26. https://doi.org/10.1090/btran/10

work page doi:10.1090/btran/10 2016

[20] [20]

(2015).Introduction to Analytic and Probabilistic Number Theory (3rd ed.)

Tenenbaum, G. (2015).Introduction to Analytic and Probabilistic Number Theory (3rd ed.). American Mathematical Society

work page 2015

[21] [21]

Torabi, H., & Fatehizadeh, A. (2021). A generalization of the Erdős-Sierpiński con- jecture.Journal of New Researches in Mathematics, 6(28), 75–82

work page 2021

[22] [22]

Weingartner, A. (2011). On the solutions ofσ(n) =σ(n+k).Journal of Integer Sequences, 14, 1–7

work page 2011

[23] [23]

Zumkeller, R. (2003). Sequence A083207. In N. J. A. Sloane (Ed.),The On-Line Encyclopedia of Integer Sequences. Published electronically athttps://oeis.org. 12

work page 2003