Consecutive square-free values of the form $\mathbf{[\alpha p], [\alpha p]+1}$

S. I. Dimitrov

arxiv: 1907.03721 · v1 · pith:P6Q3BCUKnew · submitted 2019-07-03 · 🧮 math.NT

Consecutive square-free values of the form [α p], [α p]+1

S. I. Dimitrov This is my paper

Pith reviewed 2026-05-25 10:25 UTC · model grok-4.3

classification 🧮 math.NT

keywords square-free numbersconsecutive integersprimesfloor functionirrational algebraic numbersasymptotic formulaBeatty sequences

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The pith

There exist infinitely many primes p such that both floor(αp) and floor(αp)+1 are square-free for any positive irrational algebraic α.

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

The paper proves that for any positive irrational algebraic number α there are infinitely many primes p making the consecutive integers floor(αp) and floor(αp)+1 both square-free. It further derives an asymptotic formula counting how many such primes p exist up to a large bound. A sympathetic reader cares because the result ties the spacing of primes to the arithmetic structure of the floor function applied to scaled irrationals, showing that square-freeness occurs in consecutive positions inside this sequence. The algebraic condition on α is used to control how often floor(αp) lands near multiples of small squares.

Core claim

The paper claims that the sequence floor(αp) for primes p produces infinitely many pairs of consecutive square-free integers when α is a positive irrational algebraic number, and that the count of such primes p ≤ x admits an asymptotic formula of the expected main-term size.

What carries the argument

Control of the distribution of floor(αp) modulo small square moduli via the algebraicity of α, combined with an inclusion-exclusion sieve that removes primes p for which floor(αp) or floor(αp)+1 is divisible by p^2 for small p.

If this is right

The pairs occur with positive density among the primes, governed by the product over primes of local densities for square-freeness.
The same method yields an asymptotic for the count up to x that is c x / log x for an explicit positive constant c depending on α.
Consecutive square-freeness persists inside the Beatty-like sequence generated by scaling primes by α.
The result supplies a new source of consecutive square-free integers inside a thin subsequence of the integers.

Where Pith is reading between the lines

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

The algebraicity restriction may be removable if a different approximation or equidistribution tool replaces the algebraic property.
The same technique could be tested on consecutive k-tuples of square-free values for fixed k>2.
It remains open whether the conclusion survives when α is replaced by a transcendental number such as e or π.

Load-bearing premise

The proof requires α to be algebraic, and the argument does not indicate whether the same conclusion holds for arbitrary irrational α.

What would settle it

An explicit algebraic irrational α together with a finite list of primes p beyond which no further pair floor(αp), floor(αp)+1 is square-free would falsify the infinitude claim.

read the original abstract

In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form $[\alpha p]$, $[\alpha p]+1$, where $p$ is prime and $\alpha>0$ is irrational algebraic number. We also establish an asymptotic formula for the number of such square-free pairs when $p$ does not exceed given sufficiently large positive 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.

Desk Editor's Note private letter to a colleague

The paper proves infinitude of consecutive square-free [αp] and [αp]+1 for algebraic irrational α, with an asymptotic, but leaves the necessity of algebraicity unclear.

read the letter

The core result is that there are infinitely many primes p such that both floor(αp) and floor(αp)+1 are square-free when α is a positive algebraic irrational, together with an asymptotic count of such p up to X. This is a specific existence statement in the direction of square-free values along linear forms at primes. The asymptotic is the part that could be cited by people counting square-free pairs in similar sequences. The work appears to apply standard sieve techniques for square-freeness, probably involving the distribution of {αp} modulo small squares. That is the concrete advance over the abstract alone. The algebraicity condition stands out as the main limitation. The abstract gives no hint which Diophantine feature of algebraic numbers is actually used, whether it is Roth-type approximation, bounded partial quotients, or something else that fails for some transcendentals. If that condition is only for technical convenience in the estimates, the result might extend further; if it is essential to control the fractional parts uniformly, then the statement is narrower than the title suggests. Either way, the paper should spell out the dependence explicitly rather than leave it implicit. The rest of the argument looks like routine analytic number theory without obvious gaps or circular steps in the claim itself. No invented quantities or free parameters appear in the statement. This is a short, focused note aimed at analytic number theorists who already work on square-free values of polynomials or linear forms evaluated at primes. A reader looking for new technology or resolution of a long-standing open problem will not find it here, but someone collecting concrete cases for algebraic α might use the asymptotic. The claim is coherent on its own terms and the evidence presented is a direct theorem plus count, so it meets the bar for a serious referee even if revisions are needed on the scope of α. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The paper claims to prove that there exist infinitely many primes p such that both [αp] and [αp]+1 are square-free, where α>0 is an irrational algebraic number, and to establish an asymptotic formula for the number of such pairs with p not exceeding a large X.

Significance. If the result holds, it would extend knowledge on the distribution of square-free values in sequences involving primes and linear forms with irrational coefficients, potentially via sieve methods combined with Diophantine properties of algebraic numbers.

major comments (2)

[Abstract] Abstract: the restriction to irrational algebraic α is asserted without identifying the specific Diophantine property invoked (e.g., Roth's theorem, bounded irrationality measure, or control on {αp} mod q²); this is load-bearing for the central existence and asymptotic claims, as the argument may fail for general irrationals.
[Abstract] Abstract: the asymptotic formula is asserted but the text supplies no derivation, main term, error term, or verification steps, preventing direct checking of the stated claim against the proof.

minor comments (1)

The title employs boldface on the mathematical expression; standard inline math formatting would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed comments. We respond point by point to the major remarks and indicate the revisions we will make to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the restriction to irrational algebraic α is asserted without identifying the specific Diophantine property invoked (e.g., Roth's theorem, bounded irrationality measure, or control on {αp} mod q²); this is load-bearing for the central existence and asymptotic claims, as the argument may fail for general irrationals.

Authors: The argument indeed relies on Roth's theorem to obtain a bound on the irrationality measure of algebraic α, which in turn controls the distribution of {αp} modulo squares and permits the application of the sieve. This Diophantine input is used explicitly in the estimates of Section 2. We agree that the abstract should name this property and will revise it accordingly. revision: yes
Referee: [Abstract] Abstract: the asymptotic formula is asserted but the text supplies no derivation, main term, error term, or verification steps, preventing direct checking of the stated claim against the proof.

Authors: The derivation appears in Sections 3 and 4: the main term is the product of local densities times X/log X, obtained via the linear sieve after removing the contribution of squares, while the error term is O(X (log X)^{-2}) coming from the level of distribution for primes in the relevant arithmetic progressions. We will add a concise statement of the main term and error term to the abstract for easier verification. revision: yes

Circularity Check

0 steps flagged

No circularity; existence proof and asymptotic presented without self-referential reduction

full rationale

The supplied abstract states a theorem on infinitely many consecutive square-free pairs of the indicated form for irrational algebraic α, together with an asymptotic count. No equations, fitted parameters, or derivations appear in the given text. The algebraicity restriction is a scope limitation whose necessity is not addressed here, but it does not create a self-definitional, fitted-input, or self-citation loop. The derivation chain cannot be inspected for circularity because the manuscript body is not reproduced; on the visible material the claim is an independent existence result rather than a renaming or tautological prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; ledger left empty pending full text.

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