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arxiv: 2604.27490 · v1 · submitted 2026-04-30 · 🧮 math.NT

On the difference between perfect powers and integral S-units

Yann Bugeaud This is my paper

Pith reviewed 2026-05-07 09:41 UTC · model grok-4.3

classification 🧮 math.NT
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The pith

Effective lower bounds show perfect powers and fixed-prime products must differ by amounts that grow with the power size.

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

The paper aims to prove that when z is coprime to a fixed set of primes q1 through qt, the absolute value of z raised to d minus the product of those primes raised to some powers cannot be too small. Specifically, this difference and the largest prime factor in it both tend to infinity as z to the d grows larger, for exponents d at least 2. These lower bounds are effective. A reader would care because this controls the possible closeness between perfect powers and numbers with restricted prime factors, which is a central concern in number theory for understanding Diophantine equations.

Core claim

Let q1, …, qt be distinct prime numbers. Let a1, …, at be nonnegative integers. We establish effective lower bounds for |z^d − q1^{a1}…qt^{at}| and for its greatest prime factor, which tend to infinity with z^d, where z is a positive integer coprime with q1…qt and d≥2 is an integer.

What carries the argument

Application of effective lower bounds for linear forms in logarithms to the logarithmic form arising from z^d being close to the S-unit product.

Load-bearing premise

The derivation assumes access to effective versions of lower bounds for linear forms in logarithms that apply uniformly under the coprimeness condition between z and the fixed primes.

What would settle it

Discovering infinitely many instances where z is coprime to the q_i, d >=2, z^d grows, but |z^d - S-unit| remains below some fixed bound independent of z^d would falsify the claim.

read the original abstract

Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers. We establish effective lower bounds for $|z^d - q_1^{a_1} \ldots q_t^{a_t}|$ and for its greatest prime factor, which tend to infinity with $z^d$, where $z$ is a positive integer coprime with $q_1 \ldots q_t$ and $d \ge 2$ is an integer.

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

1 major / 1 minor

Summary. The paper claims to establish effective lower bounds for |z^d - q1^{a1} ⋯ qt^{at}| (with z coprime to the fixed primes q1,...,qt and d≥2) that tend to infinity as z^d grows, together with lower bounds on the greatest prime factor of this difference that likewise tend to infinity.

Significance. If the effective constants are correctly derived, the result supplies quantitative Diophantine information on the separation between perfect powers and S-units of fixed support. Such bounds are useful for applications to S-unit equations and for showing that certain Diophantine inequalities have only finitely many solutions; the reliance on established effective versions of linear forms in logarithms is appropriate and the coprimeness hypothesis ensures the relevant linear form is non-vanishing.

major comments (1)
  1. The claim of effectiveness requires an explicit invocation of a specific effective lower bound for linear forms in logarithms (e.g., a version of Baker's theorem with explicit constants). The manuscript does not state which theorem is applied nor how the resulting constants depend on d and the fixed primes q_i; this dependence is load-bearing for the assertion that the bounds are effective and tend to infinity with z^d. (Main theorem and its proof.)
minor comments (1)
  1. Notation for the S-unit u = q1^{a1}⋯qt^{at} is introduced in the abstract but the dependence of the lower bounds on the fixed data (t, q_i, d) should be stated more explicitly in the theorem statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive suggestion regarding the explicitness of our appeal to linear forms in logarithms. We agree that greater precision on this point will strengthen the presentation of the effectiveness of our bounds.

read point-by-point responses

  1. Referee: The claim of effectiveness requires an explicit invocation of a specific effective lower bound for linear forms in logarithms (e.g., a version of Baker's theorem with explicit constants). The manuscript does not state which theorem is applied nor how the resulting constants depend on d and the fixed primes q_i; this dependence is load-bearing for the assertion that the bounds are effective and tend to infinity with z^d. (Main theorem and its proof.)

    Authors: We agree that the manuscript would benefit from an explicit citation to a specific effective theorem on linear forms in logarithms together with a brief indication of how the constants depend on d and the fixed primes q_1,...,q_t. In the revised version we will add such a reference (for instance, to an explicit form of Matveev's theorem or an equivalent result with known dependence on the number of logarithms and the heights) and explain, in the proof of the main theorem, how the lower bound for the relevant linear form in logarithms yields effective constants that grow with z^d. This clarification does not alter the argument but makes the effectiveness fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external Baker-type theorems

full rationale

The paper derives effective lower bounds on |z^d - S-unit| and its largest prime factor by substituting the coprimeness hypothesis into known explicit inequalities for linear forms in logarithms. These Baker-type results are cited as independent external input (with fixed number of logs and uniform height bounds), not fitted to the target difference or derived from prior self-citations by the same author. No equation reduces the claimed lower bound to a reparametrization of the input data, and the coprimeness condition serves only to guarantee the linear form is nonzero, which is already accommodated in the cited theorems. The central claims therefore remain non-circular and rest on externally verifiable transcendental number theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard results from transcendental number theory without new free parameters or invented entities.

axioms (1)
  • standard math Effective lower bounds for linear forms in logarithms (Baker's theorem and variants)
    Invoked to obtain explicit constants in the Diophantine lower bounds.

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Reference graph

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