pith. sign in

arxiv: 2604.15157 · v1 · submitted 2026-04-16 · 🧮 math.NT

Another factor of integer polynomials with minimal integrals

Alice Bazzanella , Carlo Sanna This is my paper

Pith reviewed 2026-05-10 09:54 UTC · model grok-4.3

classification 🧮 math.NT
keywords minimal integral polynomialsinteger polynomialsdivisibility in Z[x]prime number sumspolynomial factors over [0,1]number theory
0
0 comments X

The pith

There exist infinitely many minimal-integral integer polynomials of degree less than N that are divisible by (x^3 (1-x)^2) raised to floor(N/6).

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

The paper establishes that the set S_N of integer polynomials with degree below N and the smallest possible positive integral over [0,1] contains infinitely many members divisible by a specific higher-power factor in Z[x]. This factor is x^3(1-x)^2 to the power floor(N/6). A sympathetic reader cares because every such polynomial encodes information about primes up to N through the exact identity relating the sum of log p over prime powers to the negative log of its integral. The new result replaces an earlier weaker factor of x(1-x) to the power floor(N/3) with a stronger one that still preserves the minimal-integral condition for infinitely many choices.

Core claim

The main result states that there exist infinitely many polynomials P in S_N such that (x^3 (1-x)^2) raised to floor(N/6) divides P(x) in Z[x]. This improves the prior claim that used the lower-degree base (x(1-x)) raised to floor(N/3). The relation sum over p^m <= N of log p equals -log of the integral from 0 to 1 of P(x) dx holds for every P in S_N and supplies the motivation for studying the common factors these polynomials must share.

What carries the argument

The set S_N of degree-<N integer polynomials achieving the minimal positive integral over [0,1], together with the explicit divisibility requirement by the polynomial factor (x^3(1-x)^2)^floor(N/6).

If this is right

  • The common factor shared by polynomials in S_N has higher total degree than previously known.
  • The relation between the prime-power sum and the integral of P continues to hold for these newly identified members of S_N.
  • The same infinitude statement applies uniformly for every positive integer N.
  • The result supplies an explicit algebraic constraint that any candidate minimal polynomial must satisfy.

Where Pith is reading between the lines

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

  • The stronger factor may allow sharper explicit formulas or bounds when the prime-sum identity is used to estimate the distribution of primes.
  • Analogous divisibility statements could be sought for minimal polynomials defined over other intervals or with different positivity constraints.
  • Direct computation for moderate N would reveal whether the exponent floor(N/6) is optimal or can be increased further.

Load-bearing premise

The specific constructions that force the higher-power divisibility still produce polynomials whose integrals over [0,1] equal the global minimum value defining S_N.

What would settle it

For any fixed N, explicitly list all integer polynomials of degree less than N that are multiples of (x^3(1-x)^2)^floor(N/6) and compute their integrals; if the smallest integral among them exceeds the known minimum for S_N, the existence claim fails for that N.

read the original abstract

Let $N$ be a positive integer and let $S_N$ be the set of polynomials with integer coefficients, degree less than $N$, and minimal positive integral over $[0,1]$. D. Bazzanella initiated the study of $S_N$ because of its relation to the distribution of prime numbers. Indeed, it is possible to prove that $\sum_{p^m \leq N} \log p = -\log \int_0^1 P(x) \mathrm{d} x$ for every $P \in S_N$, where the sum runs over prime numbers $p$ and positive integers $m$ such that $p^m \leq N$. For each real number $t$, let $\lfloor t \rfloor$ denote the maximal integer not exceeding $t$. The main result of this paper states that there exist infinitely many polynomials $P \in S_N$ such that $\big(x^3(1 - x)^2\big)^{\lfloor N / 6 \rfloor}$ divides $P(x)$ in $\mathbb{Z}[x]$. This improves upon a similar result of Sanna, who proved the same claim but with the lower-degree polynomial $\big(x(1-x)\big)^{\lfloor N / 3 \rfloor}$ in place of $\big(x^3(1 - x)^2\big)^{\lfloor N / 6 \rfloor}$.

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.

Referee Report

1 major / 2 minor

Summary. The paper defines S_N as the set of polynomials P in Z[x] with deg(P) < N and minimal positive integral over [0,1] equal to 1/L with L = lcm[1..N]. It proves there exist infinitely many such P divisible in Z[x] by the factor (x^3(1-x)^2)^{floor(N/6)}, improving Sanna's analogous result that used the weaker factor (x(1-x))^{floor(N/3)}.

Significance. If the result holds, it strengthens the structural description of S_N by exhibiting a larger explicit factor compatible with the minimal-integral lattice condition. Given the link between elements of S_N and the Chebyshev function via sum_{p^m <=N} log p = -log int_0^1 P(x) dx, the improved divisibility may facilitate constructions with additional arithmetic properties or refined estimates in prime-distribution applications.

major comments (1)
  1. [Main result / proof of Theorem 1] The existence reduces to showing that the Z-module generated by the beta integrals B(3k+m+1, 2k+1) for m=0 to floor(N/6)-1 equals (1/L)Z when the quotient polynomial Q has deg < N-5k. The manuscript must supply a general argument for this module equality rather than verification only for small N (e.g., N=6,7,8,9,12).
minor comments (2)
  1. [Abstract] The abstract states the main result but omits an explicit sentence identifying the minimal integral value as 1/L; adding this would make the definition of S_N self-contained for readers unfamiliar with the prior literature.
  2. [Introduction] Notation for the floor function and the precise statement of divisibility in Z[x] are standard, but a brief reminder in the introduction that all coefficients remain integers after division would aid clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the detailed comment on the proof of the main result. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Main result / proof of Theorem 1] The existence reduces to showing that the Z-module generated by the beta integrals B(3k+m+1, 2k+1) for m=0 to floor(N/6)-1 equals (1/L)Z when the quotient polynomial Q has deg < N-5k. The manuscript must supply a general argument for this module equality rather than verification only for small N (e.g., N=6,7,8,9,12).

    Authors: We agree that the current version of the manuscript verifies the module equality only for small N and that a general argument is required for the proof to hold for arbitrary N. In the revised manuscript we will supply a complete general proof that the Z-module generated by the indicated beta integrals equals (1/L)Z. The argument proceeds by expressing the integrals in terms of the standard monomial basis of polynomials of degree less than 5k, using the relation between the beta function and the least common multiple L, and showing that the minimal positive linear combination is exactly 1/L by an inductive construction on the exponent k together with the known integrality properties of the binomial coefficients appearing in the change-of-basis matrix. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines S_N directly from the minimal positive integral condition over [0,1] together with integer coefficients and degree bound. The main theorem asserts existence of infinitely many elements of S_N divisible by the stated factor. This is compatible with the definition because the integral condition is a single linear equation on coefficients whose solvability over Z follows from explicit beta-function evaluations, without any fitted parameter or self-referential closure. The reference to Sanna's weaker result is purely comparative and not invoked as a premise or uniqueness theorem for the new construction. No step equates a derived quantity to its own input by definition, renames a known pattern, or relies on a self-citation chain for the central claim. The argument therefore stands on the explicit linear algebra of the integral constraint.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard algebraic and analytic structures without introducing new free parameters or entities.

axioms (4)
  • standard math Z[x] is a ring in which divisibility is well-defined
    Basic fact of polynomial rings over integers
  • standard math The integral from 0 to 1 of a polynomial is a well-defined real number
    Standard calculus
  • domain assumption S_N is non-empty for every positive integer N
    Implicit in the definition and prior work mentioned
  • domain assumption The prime-sum formula holds for every P in S_N
    Stated as provable in the abstract

pith-pipeline@v0.9.0 · 5546 in / 1433 out tokens · 43903 ms · 2026-05-10T09:54:04.003888+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    Aparicio Bernardo,On the asymptotic structure of the polynomials of minimal Diophantic deviation from zero, J

    E. Aparicio Bernardo,On the asymptotic structure of the polynomials of minimal Diophantic deviation from zero, J. Approx. Theory55(1988), no. 3, 270–278. MR 968933

    work page 1988

  2. [2]

    R. C. Baker, G. Harman, and J. Pintz,The difference between consecutive primes. II, Proc. London Math. Soc. (3)83(2001), no. 3, 532–562. MR 1851081

    work page 2001

  3. [3]

    Bazzanella,A note on integer polynomials with small integrals, Acta Math

    D. Bazzanella,A note on integer polynomials with small integrals, Acta Math. Hungar.141(2013), no. 4, 320–328. MR 3122290

    work page 2013

  4. [4]

    Bazzanella,A note on integer polynomials with small integrals

    D. Bazzanella,A note on integer polynomials with small integrals. II, Acta Math. Hungar.149(2016), no. 1, 71–81. MR 3498948

    work page 2016

  5. [5]

    Bazzanella,Integer polynomials with small integrals, Riv

    D. Bazzanella,Integer polynomials with small integrals, Riv. Math. Univ. Parma (N.S.)7(2016), no. 1, 165–179. MR 3675407

    work page 2016

  6. [6]

    Bordell` es,Arithmetic tales, advanced ed., Universitext, Springer, Cham, [2020]©2020, Translated by V´ eronique Bordell` es

    O. Bordell` es,Arithmetic tales, advanced ed., Universitext, Springer, Cham, [2020]©2020, Translated by V´ eronique Bordell` es. MR 4199098

    work page 2020

  7. [7]

    Borwein and T

    P. Borwein and T. Erd´ elyi,The integer Chebyshev problem, Math. Comp.65(1996), no. 214, 661–681. MR 1333305

    work page 1996

  8. [8]

    P. L. Chebyshev,Collected Works. Vol. I. Theory of Numbers, Akad. Nauk SSSR, Moscow-Leningrad,

    work page

  9. [9]

    Granville,Number theory revealed: an introduction, American Mathematical Society, Providence, RI, 2019

    A. Granville,Number theory revealed: an introduction, American Mathematical Society, Providence, RI, 2019. MR 3970983

    work page 2019

  10. [10]

    E. E. Kummer,¨ uber die Erg¨ anzungss¨ atze zu den allgemeinen Reciprocit¨ atsgesetzen, J. Reine Angew. Math.44(1852), 93–146. MR 1578793

    work page

  11. [11]

    H. L. Montgomery,Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543

    work page 1994

  12. [12]

    Nair,A new method in elementary prime number theory, J

    M. Nair,A new method in elementary prime number theory, J. London Math. Soc. (2)25(1982), no. 3, 385–391. MR 657495

    work page 1982

  13. [13]

    Nair,On Chebyshev-type inequalities for primes, Amer

    M. Nair,On Chebyshev-type inequalities for primes, Amer. Math. Monthly89(1982), no. 2, 126–129. MR 643279

    work page 1982

  14. [14]

    I. E. Pritsker,Small polynomials with integer coefficients, J. Anal. Math.96(2005), 151–190. MR 2177184

    work page 2005

  15. [15]

    " ]; If [ ris == {} , Print [

    C. Sanna,A factor of integer polynomials with minimal integrals, J. Th´ eor. Nombres Bordeaux29 (2017), no. 2, 637–646. MR 3682482 Department of Mathematical Sciences, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy Email address:carlo.sanna@polito.it ANOTHER F ACTOR OF INTEGER POLYNOMIALS WITH MINIMAL INTEGRALS 9 Appendix The follo...

    work page 2017