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On the greatest prime factor and uniform equidistribution of quadratic polynomials

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Infinitely often, the largest prime factor of $n^2+h$ exceeds $n^{1.312}$, unconditionally for $h=1$.

desk verdict Unconditional n^{1.312} for the greatest prime factor of n^2+1 is real, and the h-uniform Type I/II estimates are a genuine advance; the main caveat is that the whole proof rests on an unproved companion-paper theorem, plus a small typo in a quartic form. read the letter →

arxiv 2505.00493 v2 pith:Y2YTOGH3 submitted 2025-05-01 math.NT

Lasse Grimmelt , Jori Merikoski This is my paper · ORCID
classification math.NT MSC 11N3211N75
keywords greatestprimefactorquadraticpolynomialsTypeIandIIestimatesequidistributionofrootscongruencesautomorphickernelsspectralgapdivisorproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes lower bounds on the largest prime factor of the quadratic polynomial $n^2+h$ that were previously out of reach without the optimal spectral-gap conjecture for hyperbolic surfaces. The headline result is that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ for infinitely many $n$; when $h=1$ this is unconditional, and for larger shifts $h\le n^{1+o(1)}$ it holds under a natural hypothesis on the average count $\varrho_{a,h}(p)$ of roots modulo primes. The same family of estimates yields uniformity in $h$ for the equidistribution of roots $a\nu^2+h\equiv0\pmod p$ among primes, and a divisor-type estimate for $ax^2+by^3$ that feeds into a conditional asymptotic for primes of that shape. The proof removes the dependence on the optimal spectral-gap conjecture: any fixed spectral gap $\theta<1/2$ would give the same strength.

What carries the argument

The engine is the weighted average discrepancy $\langle\alpha_1|\Delta_q F|\alpha_2\rangle$: for a test function $F$ on $\mathrm{SL}_2(\mathbb{R})$, the difference between the automorphic kernel sum $\sum_{\gamma\in\Gamma_0(q)}F(\tau_1^{-1}\gamma\tau_2)$ and its Haar-average main term, paired against two linear functionals $\alpha_1,\alpha_2$. The imported Theorem 2.1 bounds a weighted version of this quantity by $(X/Y)^{1/2}Z_0^\theta$ times the square root of two kernel self-energies $\langle\alpha_i|\Delta_q k_i|\alpha_i\rangle$; the paper's contribution is to bound those self-energies sharply, over determinant-$h$ point functionals by diagonal/off-diagonal counting and over lower-triangular orbit functionals by splitting into diagonal, pseudo-diagonal, and off-diagonal pieces. The counting uses the fact that after Cauchy-Schwarz the congruence conditions take the shape $F(g_1,g_2)\equiv0\pmod d$ for a quartic form $F$, whose non-zero values can be absorbed by a divisor bound, while the zero set is sparse enough to control.

What would settle it

Fix the simplest case $h=1$, $a=1$, modulus $d=1$, choose a fixed smooth bump $F$ on $\mathrm{SL}_2(\mathbb{R})$ supported near the identity, and directly evaluate the finite sum $\sum_{k\sim K}\sum_{\ell^2+1\equiv0\pmod k}\psi(\ell/X)$ against its Poisson main term; compare the discrepancy with the bound that Theorem 2.1 would give for $\langle I|\Delta_1 F|\alpha_{1,1}\rangle$ after optimizing the auxiliary parameters, and if the ratio grows faster than $X^{o(1)}$ at any $K,X$ in the claimed range, the imported engine fails and the main theorems lose their foundation.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that a weighted average discrepancy of $\mathrm{SL}_2(\mathbb{R})$ automorphic kernels, imported as a theorem from the authors' companion work, can be turned into Type I and Type II estimates for the distribution of solutions to $a\ell^2+h\equiv0\pmod k$ that are uniform in $h$ up to $h\le X^{1+o(1)}$ and non-trivial for any spectral gap $\theta<1/2$. The proof encodes each solution by a symmetric integer matrix of determinant $h$, lets the congruence subgroup $\Gamma_0(d)$ act on these matrices, and bounds the resulting averages over determinant-$h$ point sets and lower-triangular orbits by explicit counting with divisor bounds. The decisive feature is that the divisor-switching symmetry is preserved in physical space, which is what removes the need for the optimal spectral-gap conjecture; plugging these estimates into the sieve arguments gives the exponent $1.312$, the uniformity in $h$, and the equidistribution statement.

Load-bearing premise

All main estimates inherit their power from a weighted-average-discrepancy theorem imported verbatim from the authors' companion work, and the present paper proves neither that theorem nor that all of its hypotheses hold in these applications; if that imported bound is wrong or inapplicable at any parameter choice used here, the Type I and Type II estimates and the theorems built on them collapse.

Editorial extensions

If this is right

  • For $h=1$, the greatest prime factor of $n^2+1$ exceeds $n^{1.312}$ infinitely often, with no unproved spectral input; this directly improves the known unconditional exponent $1.3$.
  • For square-free $h$ with $1\le h\le X^{1+\varepsilon}$, the same lower bound $>X^{1.312}$ holds under the $\varrho_{a,h}$ hypothesis; for $h\le X^{\varepsilon^2}$ the hypothesis is provable unconditionally from the classical zero-free region.
  • The roots $\nu$ modulo primes $p\le X$ of $a\nu^2+h\equiv0\pmod p$ are equidistributed in intervals $(\alpha p,\beta p]$, uniformly in square-free $h\le X^{1+o(1)}$, with the $\varrho_{a,h}$ average as the main term.
  • The Type I/II estimates are non-trivial for any spectral gap $\theta<1/2$, so any unconditional improvement of the gap translates directly into wider uniformity or larger exponents.
  • The divisor estimate for $ax^2+by^3$ supplies the missing input of a conditional asymptotic count of primes of that shape under a Hecke-eigenvalue cancellation conjecture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the method only needs any fixed spectral gap $\theta<1/2$, the same physical divisor-switching may transfer to other congruence subgroups or higher-degree binary forms; the natural test is whether the lower-triangular orbit counting generalizes to a higher-rank analogue.
  • The $\varrho_{a,h}$ hypothesis is essentially a statement that $an^2+h$ has no excess of small prime factors; a deeper zero-free-region argument than the one sketched might make the uniformity in $h$ unconditional across all square-free $h\le X^{1+\varepsilon}$.
  • For shifts near the upper limit $h\approx X^2$ the paper notes the estimates remain non-trivial but the exponent worsens to a value $\varpi(h)\to1$; optimizing the auxiliary parameters there is a direct way to extend the range.
  • The uniformity in $h$ may be useful beyond roots of quadratics: the same automorphic-kernel discrepancy bounds could apply to other sequences defined by congruence conditions on symmetric matrices, such as representation numbers of binary quadratic forms in short intervals.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves new estimates for the greatest prime factor of an^2+h and for the equidistribution of the roots of aν^2+h≡0 mod p. The main result (Theorem 1.1) gives P(an^2+h)>X^{1.312} for some n∈[X,2X], for square-free h≤X^{1+ε} and a≤X^ε, under an averaged real-character hypothesis on ϱ_{a,h}(p); for h=1 the result is unconditional and reproduces, without Selberg eigenvalues, the range previously known only under Selberg's eigenvalue conjecture. The proof is built on two uniform estimates, Theorem 1.4 (Type I) and Theorem 1.5 (Type II), which are derived from a weighted-average discrepancy bound for SL2(R) automorphic kernels imported from the companion paper [5], together with parametrizations of symmetric determinant-h matrices by Heegner points and lower-triangular orbits. Section 8 gives a variant of the divisor problem for ax^2+by^3, stated as Proposition 8.2 and used in the conditional work [9] on primes of that shape.

Significance. If the imported Theorem 2.1 is correct and its hypotheses are satisfied in the applications, the paper is a substantial advance: it removes the Selberg-eigenvalue obstruction from the n^2+1 greatest-prime-factor problem, obtains exponent 1.312 unconditionally for fixed h, and extends uniform quantitative estimates for roots of quadratic congruences to h≤X^{1+o(1)} under a cleanly stated character hypothesis. The paper also gives a workable template for extending Duke-Friedlander-Iwaniec to large shifts. Strengths worth emphasizing: the exponent 1.312 is proved rather than fitted; the external ϱ_{a,h} hypothesis is stated explicitly and separated from the automorphic argument; and the auxiliary bounds in Propositions 4.1, 4.2, and 8.2 are argued in detail and are potentially reusable. The main caveat is that the central spectral engine is entirely contained in the companion paper [5], so the significance of the present paper is conditional on the correctness and availability of that result.

major comments (2)
  1. [§4.1.2, quartic form F] The congruence condition (4.1) for τ=(* *; c0 d0) implies u≡λ(c0^2,2c0d0,d0^2) mod q, and hence u2^2≡4u1u3 mod q. The displayed form F(w1,z2)=u2^2−2u1u3 is therefore a factor-2 error: the stated implication F(w1,z2)≡0 mod q does not follow. This off-diagonal congruence is what lets the q-sum in Proposition 4.1 be absorbed by a divisor bound, and Proposition 4.1 enters both the Type I and Type II bounds, so the error is load-bearing. It appears local: replacing 2 by 4 restores the argument, and the subsequent non-vanishing statement works for the corrected form.
  2. [§2, Theorem 2.1] Theorem 2.1 is imported verbatim as Theorem 8.1 of the companion preprint [5] and is the engine behind Theorems 1.4, 1.5, and Proposition 8.2; no proof or even a reduction to known spectral results is given here. The applications in §5 and §6 rely on the theorem being valid for the specific compactly supported functionals α_{d,a,h} and β_{s,n1,n2}, with Hecke operators T_{h,1} and with q=ad or q=asn1n2, and on the q^{o(1)} and θ-power uniformity being uniform in the parameters. If Theorem 8.1 of [5] carries any hidden restriction not met in these applications, the main results collapse. The paper should either include a proof (or a detailed verification of the hypotheses) in an appendix, or state clearly that [5] has been accepted for publication and provide a version available to the referee.
minor comments (4)
  1. [§5, after (5.2)] The sentence 'By Proposition 4.2 with q=n1 and D=N0=N2=T=V=1' is unclear because q is not a variable of Proposition 4.2; the intended specialization of the parameters (presumably N1 and the modulus variable) should be stated explicitly.
  2. [§6, off-diagonal contribution] The variable r appears in expressions such as ψ(mdr/M) and mod mdn0njr^2 where the surrounding argument consistently uses t (with s=dn0t^2); this notational slip should be corrected throughout the paragraph.
  3. [§4.2.2] The opening phrase 'We note consider c′=0' should read 'We now consider c′=0'; the current wording appears to be a typographical slip.
  4. [References] Reference [5] is listed only as a 2025 preprint without an arXiv identifier or publication status; since the present paper's main results depend on it, the reference should be made fully verifiable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the headline exponent is derived through a chain of estimates whose main imported input (Theorem 2.1 from the authors' companion paper) is a stated standalone bound, not a restatement of the conclusions.

full rationale

The derivation is not circular. Theorem 1.1 is obtained from the Type I and Type II estimates (Theorems 1.4 and 1.5) via the sieve arguments of [8] and [4]; the exponent 1.312 is a consequence of the stated bounds and the Kim–Sarnak bound θ ≤ 7/64, not a fitted or pre-supposed quantity. No parameter is fitted to the quantity being predicted, and the ϱ_{a,h} hypothesis is an exterior input about character sums that is neither implied by nor used to define the greatest-prime-factor conclusion. The central import is Theorem 2.1, quoted as 'The following is Theorem 8.1 in [5]'; although this is a same-author citation and is load-bearing for Theorems 1.4, 1.5, and Proposition 8.2, it is a standalone weighted-average discrepancy bound with explicit hypotheses on q, F, Z0Z1Z2, and the functionals α1, α2,h. The applications in Sections 5 and 6 verify those hypotheses, including support, derivative bounds, X/Y > δ, and the factorization Z0Z1Z2 = Xh^{-1/2}; the paper does not use the main theorems of the present paper to prove Theorem 2.1, nor does Theorem 2.1 by construction encode the desired largest-prime-factor or equidistribution statements. A gap or unmet hypothesis in Theorem 2.1 would be a correctness risk, not circularity. Similarly, Lemma 8.3 cites [9, Section 5] for truncated Poisson summation and the Weil bound, which are standard external ingredients. The self-citations therefore do not make the derivation circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new objects. Its contributions hinge on a previously built theorem (Theorem 2.1) and on an external character-sum hypothesis for the uniform-h statement. The h=1 headline result uses only standard background and the imported theorem.

assumptions (4)
  • ad hoc to paper Theorem 2.1 (weighted average discrepancy bound for SL2(R) automorphic kernels), imported as Theorem 8.1 of [5]
    Used as the engine of Theorems 1.4 and 1.5 and Proposition 8.2. The present paper does not prove it; [5] is a preprint by the same authors.
  • domain assumption Natural hypothesis on real character sums: for all X^ε < Y < Z ≤ X^2, sum_{Y≤p<Z} (log p / p) ϱ_{a,h}(p) ≤ (1+ε) log(Z/Y) + 1/ε
    Assumed in Theorem 1.1 to get uniformity in h; for h=1 it is known, so the headline n^2+1 result does not need it.
  • standard math Known spectral gap θ≤7/64 for congruence subgroups (Kim-Sarnak)
    Used to keep the error terms nontrivial; the method only requires θ<1/2, so it is independent of future progress.
  • standard math Classical zero-free region for Dirichlet L-functions, used to establish the ϱ hypothesis for h≤X^{ε^2}
    Mentioned in Section 1; standard background.

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Pith. "Pith review of On the greatest prime factor and uniform equidistribution of quadratic polynomials." pith.science (2026). https://pith.science/paper/Y2YTOGH3

@misc{pith2026250500493,
  author       = {Pith},
  title        = {Pith review of: On the greatest prime factor and uniform equidistribution of quadratic polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2YTOGH3}},
  note         = {Machine review of arXiv:2505.00493}
}
abstract

We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq n^{1+o(1)}$ under a natural hypothesis on real characters. The same uniformity is obtained for the equidistribution of the roots of quadratic congruences modulo primes. We also prove a variant of the divisor problem for $ax^2+by^3$, which was used by the second author to give a conditional result about primes of that shape.

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Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · cited by 2 Pith papers

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