REVIEW 1 major objections 5 minor 11 references
On the largest prime factor of $n^2+1$
T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read This paper establishes that infinitely many integers $n$ have the largest prime factor of $n^2+1$ greater than $n^{1.279}$, and that assuming the eigenvalue conjecture the exponent can be raised to $1.312$.
desk verdict A legitimate record-breaker whose real value is the new Type II/Harman's-sieve combination; the numerical margin is thin but the structure is honest and sound. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is a Type II (bilinear) estimate. Defined in Proposition 4, it gives an asymptotic formula, up to $O(x^{1-\eta})$, for $\sum_{m\sim M}\sum_{n\sim N} a_m b_n \sum_{\nu^2+1\equiv0\pmod{mn}} \psi_P(mn)\log(mn)$, for divisor-bounded $a_m,b_n$ with $b_n$ square-free, provided $MN=P=x^\alpha$ and $N$ lies in one of two ranges. The proof goes through Poisson summation, Cauchy-Schwarz, and completion of an incomplete exponential sum to a linear form of Kloosterman sums, bounded with the current spectral-gap parameter $\theta=7/64$; the sieve applies this estimate to all but finitely many pieces of an iterated decomposition of $S(A(P),2\sqrt{P})$, and the leftover pieces are bounded by numerical integrals arising from the sieve.
What would settle it
Recompute the numerical constants that close the proof: the paper reports that at $\alpha=1.279$ the final coefficient is $0.997\cdots<1$ after summing $25/157+G_1+G_2+G_3+G_4+G_5-G_6$ and the linear-sieve integral. A recalculation reaching $1$ or more would destroy the contradiction that forces the theorem. A direct disproof would be infinitely many $n$ with largest prime factor $\le n^{1.279}$, which the theorem says cannot happen.
Extended reading notes
Core claim
On the paper's own terms, the central claim is Theorem 1: for infinitely many integers $n$, the largest prime factor of $n^2+1$ exceeds $n^{1.279}$. The paper's new ingredient is Proposition 4, an asymptotic formula for bilinear sums over $m\sim M$ and $n\sim N$ with $MN=P$, in which the counting term $\sum_{\nu^2+1\equiv0\pmod{mn}} 1$ is matched to its expected main term up to an error $O(x^{1-\eta})$; this holds when $N$ lies in a stated range depending on the spectral parameter $\theta=7/64$. The sieve argument decomposes the prime-counting sum $S(x,P)$ with the standard iterated identity for sifted sums, applies the bilinear formula to the pieces in range, and bounds the remaining pieces numerically; adding the ranges up to $x^{153/128}$ and the linear-sieve bound beyond it keeps the total below the required $X\log x$, forcing the existence of infinitely many $n$ with a prime factor above $n^{1.279}$. Theorem 2 is the same argument with $\theta=0$, yielding $1.312$ conditionally.
Load-bearing premise
The whole gain beyond the old exponent $1.23$ rests on a single deep external theorem: a lower bound for the smallest Laplace eigenvalue of the congruence subgroups used in the argument, written as $\lambda_1\ge 1/4-(7/64)^2$. If that bound were weakened, the exponent in Theorem 1 would drop to $1.23$; the argument does not create this assumption itself.
Editorial extensions
If this is right
- Infinitely many $n$ have $P^+(n^2+1)>n^{1.279}$, and the proof locates such $n$ through a contradiction count rather than by construction.
- Assuming the spectral eigenvalue conjecture, the same method gives infinitely many $n$ with $P^+(n^2+1)>n^{1.312}$, improving the previous conditional $1.2247$.
- The new Type II estimate extends the range in which the arithmetic of $n^2+1$ is understood from $P=x^{1+o(1)}$ to an explicit interval, and the sieve bound varies continuously from an asymptotic formula to the linear-sieve bound as $P$ grows.
- The paper's numerical constants could be pushed slightly higher by more careful optimization, so the exact limit of the method is not exhausted by the stated $1.279$.
Reading between the lines
- If the average over the auxiliary level variable $r$ could be exploited, as in the paper's Conjecture 1, the present method would already reach exponent $1.286$ with the current spectral parameter; this marks the nearest plausible next step.
- Further progress beyond $x^{153/128}$ appears to require genuinely new higher-order arithmetic information, such as asymptotic formulas for Type $I_2$ or $I_3$ sums; the paper notes that even the simplest Type $I_2$ case remains open.
- The same strategy should transfer to $n^2-d$ for non-square $d$, giving analogous records for other quadratic sequences; the paper states this expectation but does not prove it.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that the largest prime factor of n^2+1 is infinitely often greater than n^1.279 (Theorem 1), improving the previous exponent 1.2182 of de la Bretèche and Drappeau. It also proves Theorem 2: assuming Selberg's eigenvalue conjecture, the exponent can be raised to 1.312. The method follows Chebyshev and Hooley in relating the desired lower bound to upper bounds for sums over primes p∼P. The authors combine the Type I information of de la Bretèche–Drappeau (Proposition 3) with a new Type II estimate (Proposition 4) obtained from Deshouillers–Iwaniec bounds on linear forms of Kloosterman sums together with the Kim–Sarnak spectral gap θ=7/64. The sieve part uses Harman's method and Buchstab's identity to produce explicit integral-defined deficiencies G1–G6, which are then summed to obtain the final numerical inequality. The paper contains a detailed proof of the Type II estimate (Section 3), including the application of the Cauchy–Schwarz inequality, completion of sums, and the Deshouillers–Iwaniec bound.
Significance. Should the result be correct, it is a substantial improvement in a classical problem of number theory: the exponent 1.279 improves on de la Bretèche–Drappeau's 1.2182, and the conditional 1.312 improves on Deshouillers–Iwaniec's conditional 1.2247. The main innovation, a Type II estimate for the quadratic congruence n^2+1≡0 mod m with a power saving of x, is clearly presented and is of independent interest for other applications of Harman's sieve. The proof is modular and transparent: Propositions 3 and 4 are clearly isolated as arithmetic inputs, the Buchstab decompositions in Section 2.4 are explicit, and the error terms are tracked. The paper also makes the numerical code available (though in ephemeral form). The most significant residual risk is that the final numerical margin is only about 3×10^{-3} and the numerical quadratures are not certified; this is a verification gap rather than a detected flaw in the analytic argument.
major comments (1)
- [§2.4–2.5, end of Section 2.5] The conclusion of both Theorem 1 and Theorem 2 hinges on numerical inequalities with a margin of roughly 3×10^{-3}: in §2.5, 0.553361 + 4(1−2θ)∫_{153/128}^{1.279} α/(1−2θα) dα = 0.997… < 1, and in §2.6, 0.679914 + 4∫_{5/4}^{1.312} α dα = 0.997… < 1. The constants feeding these inequalities (G1<0.01745, G2<0.11478, G3<0.093754, G4<0.0057, G5<0.17877, G6>0.016329) are obtained by Python 3.7 quadrature, and the manuscript directs the reader to codepad.org links at the end of Section 2.5. No error analysis of the numerical integration is given, and a relative error in the computed constants larger than about 3×10^{-3} would overturn the theorem. Because this numerical verification is load-bearing, the revision should provide the code in a permanent repository, state the numerical method and precision, and supply either an interval-arithmetic certification or an independent evaluation of the integrals G1–G6 and the two final integrals.
minor comments (5)
- [§2.3, Lemma 7] The passage 'for (m,P(x^η))=1 we can replace ρ(m) by 1 with negligible error by equidistribution of primes in arithmetic progressions' is terse: ρ(p) equals 2 or 0 according to p≡1 or 3 mod 4, so the replacement needs an averaging argument over the ordered variables q_1,…,q_k and m. Please provide the precise statement and proof (or a reference) showing that the error is o(1) in (2.4)–(2.7) uniformly in the ranges used.
- [§2.4.1] The parenthetical '(for α < 758/733 part (ii) of Proposition 4 is stronger than (i))' is imprecise: by Remark 8 part (ii) is stronger than (i) for all α < 2671/2496 ≈ 1.0701. Please state the correct threshold or rephrase the sentence.
- [§3.8] In the sentence 'Since t ≠ 0 ≠ h1 n2 − n2 h1', the second term should read h2 n1, i.e. h1n2 − h2n1 ≠ 0, consistent with the definition of n in the same paragraph.
- [Title and Abstract] The title in the full text contains the spacing artifact 'P F ACTOR', and the arXiv abstract gives the de la Bretèche–Drappeau year as 2019 while the text and reference [1] give 2020; please unify the citations.
- [§2.5, code links] The codepad.org links at the end of Section 2.5 are ephemeral and should be replaced in the final version by a permanent repository (e.g., Zenodo or a journal-hosted supplementary file) with a versioned, runnable script.
Circularity Check
No significant circularity: the exponent 1.279 is obtained by an independent Harman-sieve optimization over external analytic inputs, not by fitting the answer.
full rationale
The paper's derivation chain is self-contained in the relevant sense: the target exponent is not used to define any of the arithmetic information. Proposition 3 (Type I) is quoted from de la Bretèche and Drappeau, Proposition 4 (Type II) is proved in Section 3 from external deep inputs, namely the Deshouillers–Iwaniec bound for linear forms of Kloosterman sums together with the Kim–Sarnak spectral gap λ1(q) ≥ 1/4 − (7/64)^2. These inputs have assumptions that do not include the largest-prime-factor conclusion, and no parameter is fitted to the desired exponent 1.279. The final numerical check is an actual inequality: the paper shows 0.553361 + 4(1−2θ)∫_{153/128}^{1.279} α/(1−2θα) dα = 0.997... < 1, and then derives the theorem by contradiction with the Chebyshev–Hooley identity. The exponent 1.279 is chosen only after this bound is computed, so the claim is not definitionally forced. There is no load-bearing self-citation: the author cites prior work of others (de la Bretèche–Drappeau, Deshouillers–Iwaniec, Kim–Sarnak, Harman, Hooley, Duke–Friedlander–Iwaniec) as external theorems, and the paper's own Type II estimate is proved independently in Section 3. Remarks 1 and 2 explicitly describe the limits of the method and the possibility of further numerical optimization, which further confirms that the numerical margin is a verification point rather than a circular device. No step in the derivation reduces, by construction or by self-citation, to its own input.
Assumptions & free parameters
assumptions (5)
- standard math Prime Number Theorem in arithmetic progressions (used to count integers coprime to P(z)).
- domain assumption Kim-Sarnak bound: λ1(q) ≥ 1/4 - (7/64)^2 for all q.
- domain assumption Deshouillers-Iwaniec bound on linear forms of Kloosterman sums, Lemma 12.
- domain assumption Type I estimate of de la Bretèche-Drappeau, Proposition 3.
- standard math Known bounds for the Buchstab function, equation (2.5).
Cite this review
Pith. "Pith review of On the largest prime factor of $n^2+1$." pith.science (2026). https://pith.science/paper/CFZQ35RU
@misc{pith2026190808816,
author = {Pith},
title = {Pith review of: On the largest prime factor of $n^2+1$},
year = {2026},
howpublished = {\url{https://pith.science/paper/CFZQ35RU}},
note = {Machine review of arXiv:1908.08816}
}
abstract
We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret\`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
Reference graph
Works this paper leans on
-
[1]
R. de la Bretèche and S. Drappeau. Niveau de répartition d es polynômes quadratiques et crible majorant pour les entiers friables. J. Eur. Math. Soc. (JEMS) , 22(5):1577–1624, 2020
work page 2020
-
[2]
J.-M. Deshouillers and H. Iwaniec. On the greatest prime factor of n2 + 1 . Ann. Inst. Fourier (Grenoble), 32(4):1–11 (1983), 1982
work page 1983
-
[3]
J.-M. Deshouillers and H. Iwaniec. Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math., 70(2):219–288, 1982/83
work page 1982
-
[4]
W. Duke, J. B. Friedlander, and H. Iwaniec. Equidistribu tion of roots of a quadratic congruence to prime moduli. Ann. of Math. (2) , 141(2):423–441, 1995
work page 1995
-
[5]
J. Friedlander and H. Iwaniec. Opera de cribro , volume 57 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2010
work page 2010
-
[6]
G. Harman. Prime-detecting sieves, volume 33 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2007
work page 2007
-
[7]
C. Hooley. On the greatest prime factor of a quadratic pol ynomial. Acta Math., 117:281–299, 1967
work page 1967
-
[8]
H. Iwaniec. Almost-primes represented by quadratic pol ynomials. Invent. Math. , 47(2):171–188, 1978
work page 1978
Show all 11 references
-
[9]
C. Jia. Almost all short intervals containing prime numb ers. Acta Arith., 76(1):21–84, 1996
1996
-
[10]
H. H. Kim. Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. J. Amer. Math. Soc. , 16(1):139–183, 2003. With appendix 1 by Dinakar Ramakrish nan and appendix 2 by Kim and Peter Sarnak
2003
-
[11]
R. J. Lemke Oliver. Almost-primes represented by quadr atic polynomials. Acta Arith., 151(3):241– 261, 2012. Department of Mathematics and Statistics, University of Turku , FI-20014 University of Turku, Finland Email address : jori.e.merikoski@utu.fi
2012
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.