Prime numbers and factorization of polynomials

Jitender Singh

arxiv: 2411.18366 · v1 · pith:WKCXVI7Tnew · submitted 2024-11-27 · 🧮 math.NT · math.AG

Prime numbers and factorization of polynomials

Jitender Singh This is my paper

Pith reviewed 2026-05-23 08:27 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords irreducible polynomialsinteger coefficientsprime factorizationirreducibility criteriabivariate polynomialsnon-Archimedean valuationspolynomial factorization

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The pith

Upper bounds on the number of irreducible factors of certain integer polynomials follow from the prime factorization of their values at large integers combined with root locations.

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

The paper seeks to establish explicit upper bounds on the number of irreducible factors over the integers for selected classes of polynomials with integer coefficients. The method relies on the prime factorization of the integer values the polynomial takes at sufficiently large arguments together with data on where the roots lie in the complex plane. When these bounds are obtained they immediately recover several classical irreducibility criteria as special cases. The same approach is carried over to bivariate polynomials over arbitrary fields by means of non-Archimedean absolute values, thereby extending earlier bivariate irreducibility statements to this wider setting.

Core claim

For certain classes of polynomials with integer coefficients, the prime factorization of the values taken at sufficiently large integers, when combined with information on the location of the roots in the complex plane, determines an explicit upper bound on the number of irreducible factors; the same valuation techniques applied to bivariate polynomials over any field produce corresponding bounds and extend known irreducibility results to that setting.

What carries the argument

Prime factorization information from polynomial evaluations at large integers together with complex root location data.

If this is right

Well-known irreducibility criteria for integer polynomials arise directly as corollaries of the bounds.
The number of irreducible factors of the polynomials in the treated classes is bounded by a quantity determined by the number of distinct prime factors of f(n) for large n, adjusted by root data.
The valuation method yields irreducibility criteria for bivariate polynomials over arbitrary fields.
Results previously obtained for bivariate polynomials by Murty and Weintraub extend to wider classes via the non-Archimedean approach.

Where Pith is reading between the lines

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

The same evaluation-plus-valuation technique might apply to polynomials over other rings where suitable absolute values exist.
Explicit numerical bounds could be computed for low-degree examples to test sharpness.
The method suggests a possible route to partial factorization algorithms that stop once the bound is reached.

Load-bearing premise

Prime factorization data from the values at large integers plus root location information is available and sufficient to produce the stated upper bounds on the number of irreducible factors.

What would settle it

A concrete polynomial with integer coefficients whose number of irreducible factors exceeds the upper bound computed from the prime factors of its values at large integers and its root locations.

read the original abstract

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we use the information about prime factorization of the values taken by such polynomials at sufficiently large integer arguments along with the information about their root location in the complex plane. Further, these techniques are extended to bivariate polynomials over arbitrary fields using non-Archimedean absolute values, yielding extensions of the irreducibility results of M. Ram Murty and S. Weintraub to bivariate polynomials.

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 gives upper bounds on irreducible factors for some integer polynomials by combining value factorizations at large integers with root locations, plus a bivariate extension over general fields; the link from data to bound is the part that needs verification.

read the letter

The central move is using prime factors of p(n) for large n together with root positions in the complex plane to cap the number of irreducible factors over Z, and recovering some classical criteria along the way. The same idea is then carried to bivariate polynomials over arbitrary fields via non-Archimedean valuations, which produces extensions of the Murty-Weintraub results. That bivariate step is the clearest addition relative to the cited work. The strategy itself is built from standard arithmetic and geometric ingredients, so the outline is coherent even if the payoff is incremental. If the arguments close cleanly, the bounds could be handy for checking specific families. The soft spot is whether the observed prime factors and root data actually force the stated upper bound. Different assignments of those primes to hypothetical factors could in principle be compatible with the same roots, so the paper has to supply an explicit reason why no such assignment exceeds the claimed count. The abstract treats the combination as sufficient, but without seeing how root locations constrain the possible degree splittings or factor distributions, that step is not automatic. This is aimed at number theorists who work on irreducibility criteria or arithmetic properties of polynomials. A specialist might get something out of the bivariate extension if the details hold. The work is coherent enough on its own terms to deserve referee time rather than a desk rejection.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to derive upper bounds on the number of irreducible factors over the integers for certain classes of polynomials by combining the prime factorization of their values at sufficiently large integer arguments with root-location data in the complex plane; these bounds are said to recover classical irreducibility criteria. The same techniques are extended to bivariate polynomials over arbitrary fields via non-Archimedean absolute values, yielding extensions of results by Murty and Weintraub.

Significance. If the central implication holds, the work would supply a uniform arithmetic-geometric method for bounding factorizations that recovers known criteria as special cases and generalizes to bivariate settings over general fields. The reliance on external prime-factorization data and root information could make the bounds practical for explicit polynomial families, though the absence of parameter-free derivations or machine-checked components limits the immediate strength of the contribution.

major comments (1)

[Abstract] Abstract, paragraph 2: the claim that prime factorization of p(n) for large n together with root-location data yields the stated upper bounds on irreducible factors requires an explicit argument that no alternative distribution of the observed prime factors among hypothetical irreducible factors is compatible with the given root locations while exceeding the bound. Different assignments of the prime-power factors to factors of varying degrees could in principle be consistent with the same root data, so the sufficiency step must be shown to rule out such possibilities for the classes considered.

minor comments (1)

[Abstract] The abstract refers to 'some classes of polynomials' and 'well known irreducibility criteria' without naming the specific families or recovered criteria; explicit statements of both would improve readability and allow immediate verification of the claimed recovery.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater explicitness in the sufficiency argument. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 2: the claim that prime factorization of p(n) for large n together with root-location data yields the stated upper bounds on irreducible factors requires an explicit argument that no alternative distribution of the observed prime factors among hypothetical irreducible factors is compatible with the given root locations while exceeding the bound. Different assignments of the prime-power factors to factors of varying degrees could in principle be consistent with the same root data, so the sufficiency step must be shown to rule out such possibilities for the classes considered.

Authors: The proofs in Sections 2–4 already contain the required argument: root-location data (via bounds on the house or Mahler measure of hypothetical factors) restrict the admissible degree sequences of the irreducible factors over Z, while the prime factorization of p(n) for large n then caps the total number of factors once the degree sequence is fixed. Any alternative assignment of the observed prime-power factors that would produce more irreducible factors necessarily violates either the root-location constraints or the integrality of the coefficients. The same logic, transferred via non-Archimedean valuations, handles the bivariate case. Nevertheless, the abstract is too terse on this point. We will revise the abstract and insert a short clarifying paragraph immediately after the statement of the main theorems to make the sufficiency step fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from independent external inputs

full rationale

The abstract states that upper bounds are obtained by combining prime factorization data of p(n) for large n with root location information in the complex plane. These are external inputs independent of the claimed bounds, with no indication of parameter fitting, self-definition, or load-bearing self-citation. The extension to bivariate cases references results by Murty and Weintraub (distinct authors). The derivation chain therefore does not reduce to its own outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned. The work relies on standard facts from number theory and algebra (prime factorization, root locations, non-Archimedean valuations).

axioms (2)

domain assumption Prime factorization of integer values of a polynomial at large arguments combined with complex root locations determines an upper bound on irreducible factors
Invoked in abstract paragraph 2 as the basis for the bounds
domain assumption Non-Archimedean absolute values extend the univariate technique to bivariate polynomials over arbitrary fields
Stated in abstract as the method for the bivariate extension

pith-pipeline@v0.9.0 · 5611 in / 1342 out tokens · 21401 ms · 2026-05-23T08:27:57.903190+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

On an irreducibilit y theorem of A

Brillhart, J., Filaseta, M., and Odlyzko, A., (1981). On an irreducibilit y theorem of A. Cohn, Canad. J. Math 33:5, 1055–1059. https://doi.org/10.4153/CJM-1981-080-0

work page doi:10.4153/cjm-1981-080-0 1981
[2]

Bonciocat A.I., Bonciocat, N.C., and Zaharescu, A., (2011). On th e irreducibility of polynomials that take a prime power value, Bulletin math´ ematique de la Soci´ et´ e des Sciences Math´ ematiques de Roumanie, Nouvelle S´ erie 54(102), No. 1, 41–54. https://www.jstor.org/stable/43679202 10

work page arXiv 2011
[3]

C., Garg, R., Singh, J

Bonciocat, N. C., Garg, R., Singh, J. (2024). Some factorization results for bivariate polynomials. Commun. Algebra https://doi.org/10.1080/00927872.2022.2377390

work page doi:10.1080/00927872.2022.2377390 2024
[4]

Sur quelques cas d ´ ırr´ educibilit´ e des polynˆ omes ´ a co- eﬃcients rationnels, Journal de Math

Dumas, G., (1906). Sur quelques cas d ´ ırr´ educibilit´ e des polynˆ omes ´ a co- eﬃcients rationnels, Journal de Math. Puers et Appl. 12, 191–258. http://www.numdam.org/item/JMPA_1906_6_2__191_0.pdf

work page 1906
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¨Uber die Irreductibilit¨ at u´ nd einige andere Eigenschaften der Gleic hung, von welcher die Theilung der ganzen Lemniscate abh¨ angt, J

Eisenstein, G., (1850). ¨Uber die Irreductibilit¨ at u´ nd einige andere Eigenschaften der Gleic hung, von welcher die Theilung der ganzen Lemniscate abh¨ angt, J. Reine Angew. Math. , 39, 160–179. http://eudml.org/doc/183299

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Valued Fields, Springer

Engler, A.J., Prestel, A., (2000). Valued Fields, Springer

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Euler, L., (1774) Extrait d’un lettre de M. Euler le pere ´ a M. Berno ulli concernant le M´ emoire imprim´ e parmi ceux de 1771, 318.Nouveaux M´ emoires de l’Acad´ emie royale des Sciences de Berlin, Histoire, pp. 35–36 (1772). https://scholarlycommons.pacific.edu/euler-works/46 1

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On an irreducibility criterion of M

Girstmair, K., (2005). On an irreducibility criterion of M. Ram Murty , Amer. Math. Monthly 112(3), 269–270. https://doi.org/10.1080/00029890.2005.11920194

work page doi:10.1080/00029890.2005.11920194 2005
[9]

Singh, J., Garg, R. (2024). Some factorization results on polyno mials having integer coeﬃcients. Commun. Algebra 52:5, 2467–2474. https://doi.org/10.1080/00927872.2023.2301530

work page doi:10.1080/00927872.2023.2301530 2024
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and Kumar, S., (2022)

Singh, J. and Kumar, S., (2022). A note on Girstmair’s irreducibilit y criterion, Bull. Aust. Math. Soc. 106, 62–66. https://doi.org/10.1017/S0004972721000861

work page doi:10.1017/s0004972721000861 2022
[11]

Kumar, S., Singh, J. (2023). A study of some recent irreducibilit y criteria for polynomials having integer coeﬃcients. arXiv Preprint: https://arxiv.org/abs/2310.02860 pp. 1–20

work page arXiv 2023
[12]

The Curious History of Fa di Bruno’s Formula

Murty, M. Ram (2002). Prime numbers and irreducible polynomials , Amer. Math. Monthly 109(5), 452–458. https://doi.org/10.1080/00029890.2002.11919872

work page doi:10.1080/00029890.2002.11919872 2002
[13]

Neue kriterien f¨ ur die irreduzibilit¨ at algebraischer gleichungen, J

Perron, O., (1907). Neue kriterien f¨ ur die irreduzibilit¨ at algebraischer gleichungen, J. Reine Angew. Math. 132, 288–307

work page 1907
[14]

and Szeg¨ o, G., (1964).Aufgaben and Lehrs¨ atze aus der Analysis, Springer-Verlag, Berlin

P´ olya, G. and Szeg¨ o, G., (1964).Aufgaben and Lehrs¨ atze aus der Analysis, Springer-Verlag, Berlin

work page 1964
[15]

Von denjenigen Moduln, welche Pote nzen von Primzahlen sind, J

Sch¨ onemann, T., (1846). Von denjenigen Moduln, welche Pote nzen von Primzahlen sind, J. Reine Angew. Math. 32, 93–105

work page
[16]

Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences , https://oeis.org

work page
[17]

Prime numbers and factorization of polynomials

Weintraub, S. H. (2013). A mild generalization of Eisenstein’s crit erion. Proc. Amer. Math. Soc. 141(4): 1159–1160. https://doi.org/10.1090/S0002-9939-2012-10880-9 (1) Department of Mathematics, Guru Nanak Dev University, Amri tsar-143005, India jitender.math@gndu.ac.in 11 This figure "ORCID.png" is available in "png" format from: http://arxiv.org/ps/241...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/s0002-9939-2012-10880-9 2013

[1] [1]

On an irreducibilit y theorem of A

Brillhart, J., Filaseta, M., and Odlyzko, A., (1981). On an irreducibilit y theorem of A. Cohn, Canad. J. Math 33:5, 1055–1059. https://doi.org/10.4153/CJM-1981-080-0

work page doi:10.4153/cjm-1981-080-0 1981

[2] [2]

Bonciocat A.I., Bonciocat, N.C., and Zaharescu, A., (2011). On th e irreducibility of polynomials that take a prime power value, Bulletin math´ ematique de la Soci´ et´ e des Sciences Math´ ematiques de Roumanie, Nouvelle S´ erie 54(102), No. 1, 41–54. https://www.jstor.org/stable/43679202 10

work page arXiv 2011

[3] [3]

C., Garg, R., Singh, J

Bonciocat, N. C., Garg, R., Singh, J. (2024). Some factorization results for bivariate polynomials. Commun. Algebra https://doi.org/10.1080/00927872.2022.2377390

work page doi:10.1080/00927872.2022.2377390 2024

[4] [4]

Sur quelques cas d ´ ırr´ educibilit´ e des polynˆ omes ´ a co- eﬃcients rationnels, Journal de Math

Dumas, G., (1906). Sur quelques cas d ´ ırr´ educibilit´ e des polynˆ omes ´ a co- eﬃcients rationnels, Journal de Math. Puers et Appl. 12, 191–258. http://www.numdam.org/item/JMPA_1906_6_2__191_0.pdf

work page 1906

[5] [5]

¨Uber die Irreductibilit¨ at u´ nd einige andere Eigenschaften der Gleic hung, von welcher die Theilung der ganzen Lemniscate abh¨ angt, J

Eisenstein, G., (1850). ¨Uber die Irreductibilit¨ at u´ nd einige andere Eigenschaften der Gleic hung, von welcher die Theilung der ganzen Lemniscate abh¨ angt, J. Reine Angew. Math. , 39, 160–179. http://eudml.org/doc/183299

work page

[6] [6]

Valued Fields, Springer

Engler, A.J., Prestel, A., (2000). Valued Fields, Springer

work page 2000

[7] [7]

Euler le pere ´ a M

Euler, L., (1774) Extrait d’un lettre de M. Euler le pere ´ a M. Berno ulli concernant le M´ emoire imprim´ e parmi ceux de 1771, 318.Nouveaux M´ emoires de l’Acad´ emie royale des Sciences de Berlin, Histoire, pp. 35–36 (1772). https://scholarlycommons.pacific.edu/euler-works/46 1

work page

[8] [8]

On an irreducibility criterion of M

Girstmair, K., (2005). On an irreducibility criterion of M. Ram Murty , Amer. Math. Monthly 112(3), 269–270. https://doi.org/10.1080/00029890.2005.11920194

work page doi:10.1080/00029890.2005.11920194 2005

[9] [9]

Singh, J., Garg, R. (2024). Some factorization results on polyno mials having integer coeﬃcients. Commun. Algebra 52:5, 2467–2474. https://doi.org/10.1080/00927872.2023.2301530

work page doi:10.1080/00927872.2023.2301530 2024

[10] [10]

and Kumar, S., (2022)

Singh, J. and Kumar, S., (2022). A note on Girstmair’s irreducibilit y criterion, Bull. Aust. Math. Soc. 106, 62–66. https://doi.org/10.1017/S0004972721000861

work page doi:10.1017/s0004972721000861 2022

[11] [11]

Kumar, S., Singh, J. (2023). A study of some recent irreducibilit y criteria for polynomials having integer coeﬃcients. arXiv Preprint: https://arxiv.org/abs/2310.02860 pp. 1–20

work page arXiv 2023

[12] [12]

The Curious History of Fa di Bruno’s Formula

Murty, M. Ram (2002). Prime numbers and irreducible polynomials , Amer. Math. Monthly 109(5), 452–458. https://doi.org/10.1080/00029890.2002.11919872

work page doi:10.1080/00029890.2002.11919872 2002

[13] [13]

Neue kriterien f¨ ur die irreduzibilit¨ at algebraischer gleichungen, J

Perron, O., (1907). Neue kriterien f¨ ur die irreduzibilit¨ at algebraischer gleichungen, J. Reine Angew. Math. 132, 288–307

work page 1907

[14] [14]

and Szeg¨ o, G., (1964).Aufgaben and Lehrs¨ atze aus der Analysis, Springer-Verlag, Berlin

P´ olya, G. and Szeg¨ o, G., (1964).Aufgaben and Lehrs¨ atze aus der Analysis, Springer-Verlag, Berlin

work page 1964

[15] [15]

Von denjenigen Moduln, welche Pote nzen von Primzahlen sind, J

Sch¨ onemann, T., (1846). Von denjenigen Moduln, welche Pote nzen von Primzahlen sind, J. Reine Angew. Math. 32, 93–105

work page

[16] [16]

Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences , https://oeis.org

work page

[17] [17]

Prime numbers and factorization of polynomials

Weintraub, S. H. (2013). A mild generalization of Eisenstein’s crit erion. Proc. Amer. Math. Soc. 141(4): 1159–1160. https://doi.org/10.1090/S0002-9939-2012-10880-9 (1) Department of Mathematics, Guru Nanak Dev University, Amri tsar-143005, India jitender.math@gndu.ac.in 11 This figure "ORCID.png" is available in "png" format from: http://arxiv.org/ps/241...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/s0002-9939-2012-10880-9 2013