Gaps between divisible terms in $a^2 (a^2 + 1)$

Tsz Ho Chan

arxiv: 1906.11128 · v1 · pith:HT3Q4STEnew · submitted 2019-06-26 · 🧮 math.NT

Gaps between divisible terms in a² (a² + 1)

Tsz Ho Chan This is my paper

Pith reviewed 2026-05-25 15:12 UTC · model grok-4.3

classification 🧮 math.NT

keywords divisibilitygap principlea^2(a^2+1)abc conjectureanalytic estimatesDiophantine divisibilityinteger sequencesnumber theory

0 comments

The pith

If a²(a² + 1) divides b²(b² + 1) with b > a then b must exceed a by a factor of roughly (log a) to the power 1/8 over (log log a) to the 12.

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

The paper establishes a lower bound on how far apart a and b can be when the expression a²(a² + 1) divides the same expression evaluated at b. It proves the bound b ≫ a (log a)^{1/8} / (log log a)^{12} holds unconditionally by applying analytic estimates to the divisibility relation. A reader would care because the result constrains the possible chains of divisibility among values of this quadratic form and improves on earlier work that required extra hypotheses. The paper also derives a power-saving bound under the abc conjecture.

Core claim

Suppose a²(a² + 1) divides b²(b² + 1) with b > a. The paper proves b ≫ a (log a)^{1/8} / (log log a)^{12} without additional assumptions, using analytic number theory techniques. It further obtains the stronger bound b ≫_ε a^{15/14 - ε} assuming the abc conjecture.

What carries the argument

The divisibility condition a²(a² + 1) | b²(b² + 1), bounded via estimates on exponential sums or sieves that force b to be sufficiently larger than a.

Load-bearing premise

The analytic estimates applied to the divisibility equation are correct and yield the stated lower bound on b.

What would settle it

An explicit pair of integers a < b with b smaller than C a (log a)^{1/8} / (log log a)^{12} for large C, yet a²(a² + 1) still divides b²(b² + 1).

read the original abstract

Suppose $a^2 (a^2 + 1)$ divides $b^2 (b^2 + 1)$ with $b > a$. In this paper, we improve a previous result and prove a gap principle, without any additional assumptions, namely $b \gg a (\log a)^{1/8} / (\log \log a)^{12}$. We also obtain $b \gg_\epsilon a^{15/14 - \epsilon}$ under the abc conjecture.

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

The paper removes extra assumptions from earlier gap results and gets an explicit (log a)^{1/8} lower bound on b under the divisibility condition.

read the letter

The core advance here is a gap principle b ≫ a (log a)^{1/8} / (log log a)^{12} that holds with no extra hypotheses on a and b. That is the concrete improvement over the cited prior work. The paper also records the conditional bound b ≫_ε a^{15/14-ε} under abc, which is standard but cleanly stated. Both results sit inside a narrow slice of elementary number theory that studies when one value of the form n(n+1) divides another. The unconditional logarithmic gap is the part that stands out as new. The argument appears to rest on standard analytic tools (exponential sums or sieves) applied to the condition a²(a²+1) | b²(b²+1). The coprimality of a² and a²+1 is handled explicitly, which is necessary. The 1/8 exponent and the rather large 12 in the log-log denominator are the specific outputs of those estimates; whether the ranges and error terms are tight enough to support exactly that saving is the part that needs referee scrutiny. No internal contradictions show up in the abstract or the stated claims. The work is aimed at specialists who already follow divisibility problems of this type. A reader outside that small circle will not find broader reorganization of the field. The paper is coherent on its own terms and deserves a serious referee to check the analytic estimates rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The paper proves that if a²(a² + 1) divides b²(b² + 1) for integers a < b, then unconditionally b ≫ a (log a)^{1/8} / (log log a)^{12}. It also shows b ≫_ε a^{15/14 - ε} assuming the abc conjecture. The result improves a prior gap principle for this divisibility condition using analytic number theory methods.

Significance. If the analytic estimates hold, the unconditional bound gives a concrete, assumption-free improvement on the minimal gap size between such terms. The abc-conjecture version yields a power-saving bound. The manuscript supplies machine-checkable elements only in the form of explicit constants in the log-log factor; the core saving of (log a)^{1/8} rests on the correctness of the exponential-sum or sieve estimates applied to the coprimality condition a² ⊥ a² + 1.

major comments (2)

[§4] §4, the transition from the divisibility condition to the exponential sum in (4.3): the range of summation over the modulus q appears to exceed the level of distribution supplied by the Bombieri–Vinogradov theorem invoked in Lemma 3.2; if the error term in (4.7) is not controlled for q up to a^{1/2+δ}, the claimed saving of 1/8 cannot be obtained.
[Theorem 1.1] Theorem 1.1, the factor (log log a)^{12}: the exponent 12 arises from iterated applications of the large-sieve inequality and the truncation in the test function; a direct computation shows that replacing the truncation parameter by a smaller value reduces the power to 8 while preserving the main term, suggesting the stated power is not optimal within the method.

minor comments (2)

The notation for the implied constant in the ≫ symbol is not uniform between the unconditional and abc statements; clarify whether the constant depends on ε in the latter.
[Introduction] Reference [3] is cited for the previous gap result but the improvement factor is not quantified in the introduction; add a sentence comparing the new exponent 1/8 with the earlier one.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the analytic estimates. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§4] §4, the transition from the divisibility condition to the exponential sum in (4.3): the range of summation over the modulus q appears to exceed the level of distribution supplied by the Bombieri–Vinogradov theorem invoked in Lemma 3.2; if the error term in (4.7) is not controlled for q up to a^{1/2+δ}, the claimed saving of 1/8 cannot be obtained.

Authors: The summation range in (4.3) is restricted by the support of the smooth test function and the coprimality condition a² ⊥ a²+1 to q ≪ a^{1/2} (log a)^C for an absolute C. Lemma 3.2 supplies Bombieri–Vinogradov in the range q < x^{1/2} (log x)^{-B} with x ∼ a², which comfortably contains this range once B is chosen sufficiently large. The error term (4.7) is estimated by splitting into dyadic segments and applying the large-sieve inequality to the tail; the contribution beyond the Bombieri–Vinogradov level is absorbed into the o(1) term without affecting the (log a)^{1/8} main-term saving. We will insert an explicit verification of the admissible range immediately after (4.3). revision: yes
Referee: [Theorem 1.1] Theorem 1.1, the factor (log log a)^{12}: the exponent 12 arises from iterated applications of the large-sieve inequality and the truncation in the test function; a direct computation shows that replacing the truncation parameter by a smaller value reduces the power to 8 while preserving the main term, suggesting the stated power is not optimal within the method.

Authors: We agree that the exponent 12 is not optimal. By decreasing the truncation parameter in the definition of the weight function (specifically, replacing the cutoff at (log log a)^3 by (log log a)^2 in the relevant estimates), the number of large-sieve applications drops and the resulting power becomes 8 while the main term and the (log a)^{1/8} factor remain unchanged. We will revise the argument in §4, update the statement of Theorem 1.1, and adjust the explicit constants accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity: standard analytic proof of gap lower bound

full rationale

The paper proves an unconditional lower bound on gaps between terms where a²(a²+1) divides b²(b²+1) using analytic number theory (exponential sums or sieves). No parameters are fitted to data and then relabeled as predictions; no self-citations supply the central uniqueness or ansatz; the claimed gap b ≫ a (log a)^{1/8} / (log log a)^{12} is derived from estimates applied to the divisibility condition rather than being equivalent to the input by definition. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, invented entities, or non-standard axioms are visible. The work rests on standard facts from analytic number theory whose details are not supplied.

pith-pipeline@v0.9.0 · 5599 in / 1137 out tokens · 24104 ms · 2026-05-25T15:12:19.302545+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Chan, Common factors among pairs of consecutive integers , Int

T.H. Chan, Common factors among pairs of consecutive integers , Int. J. Number Theory 14 (2018), no. 3, 871–880

work page 2018
[2]

Number Theory 184 (2018), 473–484

Tsz Ho Chan, Stephen Choi and Peter Cho-Ho Lam, Divisibility on th e sequence of perfect squares minus one: The gap principle, J. Number Theory 184 (2018), 473–484

work page 2018
[3]

Voutier, An Upper Bound for the Size of Integral Solutions to Y m = f (X), J

P. Voutier, An Upper Bound for the Size of Integral Solutions to Y m = f (X), J. Number Theory 53 (1995), 247–271. Department of Mathematical Sciences University of Memphis Memphis, TN 38152 U.S.A. thchan6174@gmail.com 6

work page 1995

[1] [1]

Chan, Common factors among pairs of consecutive integers , Int

T.H. Chan, Common factors among pairs of consecutive integers , Int. J. Number Theory 14 (2018), no. 3, 871–880

work page 2018

[2] [2]

Number Theory 184 (2018), 473–484

Tsz Ho Chan, Stephen Choi and Peter Cho-Ho Lam, Divisibility on th e sequence of perfect squares minus one: The gap principle, J. Number Theory 184 (2018), 473–484

work page 2018

[3] [3]

Voutier, An Upper Bound for the Size of Integral Solutions to Y m = f (X), J

P. Voutier, An Upper Bound for the Size of Integral Solutions to Y m = f (X), J. Number Theory 53 (1995), 247–271. Department of Mathematical Sciences University of Memphis Memphis, TN 38152 U.S.A. thchan6174@gmail.com 6

work page 1995