On Coverings of the Integers Associated with an Irreducibility Theorem of A. Schinzel

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open_problem_1: Asks for finite distinct-modulus coverings with arbitrarily large minimum modulus.

open_problem_2: Asks whether finitely many congruences with distinct odd moduli greater than one can cover every integer.

theorem_1: Constructs a positive-coefficient polynomial whose lacunary shifts plus d are reducible for every nonnegative exponent when four divides d.

theorem_2: Shows that a nontrivial polynomial reducibility construction for odd d would force an odd covering of the integers.

theorem_3: Gives conditions on a covering, built from the classes 2^(j-1) mod 2^j and further classes with distinct moduli, under which some positive-coefficient f makes f(x)x^n+d reducible for every nonnegative n.

theorem_4: Constructs a covering by odd composite moduli while allowing each modulus to occur at most three times.

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Michael Filaseta, On coverings of the integers associated with an irreducibility theorem of A. Schinzel, in Number Theory for the Millennium, II (Urbana, Illinois, 2000), A K Peters, Natick, Massachusetts (2002), 1--24.

The copy read for this card is a 24-page author manuscript dated 22 May 2001. Filaseta's official publication list supplies the proceedings citation above, and his seminar list records a talk with this title at the Millennial Conference on Number Theory on 23 April 2000. The result pages use the author manuscript and cite its printed page numbers, which run one behind the PDF's physical pages (the title page is unnumbered), giving the PDF page beside them; a published proceedings PDF was not compared. The manuscript prints no copyright or license line, and the author's publication list that provides it states no terms (https://people.math.sc.edu/filaseta/paperindex.html, read 2026-10-02); the term is unstated.

The introduction records two covering questions. Open Problem 1 is the arbitrarily-large minimum-modulus question now recorded as Problem 2. Open Problem 2 is exactly the distinct odd-cover question Problem 7. They are historical source statements, not evidence that either question retained its 2001 status.

The polynomial part fixes a positive integer $d$ and asks for $f\in\mathbb Z[x]$ with $f(1)\neq-d$ such that $f(x)x^n+d$ is reducible over $\mathbb Q$ for every $n\geq0$. Theorem 1 constructs such behavior with positive coefficients whenever $4\mid d$. Theorem 2 shows that for odd $d$, a nontrivial example would imply an odd covering. Theorem 3 is the bridge: it lists conditions on a covering, combining the classes $2^{j-1}\bmod 2^j$ with further classes of distinct moduli, under which a suitable positive-coefficient $f$ exists, and Theorem 1 follows by applying it to a system built from Theorem 4.

The preliminary Theorem 4 constructs a covering whose moduli are odd, greater than one, and have at least two distinct prime factors, while allowing any given modulus to occur up to three times. The repetition allowance is essential: the theorem does not produce the distinct-modulus cover asked for in Problem 7.

Compiled scope

The introduction and statements on printed pp. 1--4, Theorem 3 on printed pp. 9--10 and Theorem 4 on printed p. 12 were read clause by clause against the manuscript's page images. The six selected statements below are restated from the author manuscript read. Open Problem 3 (p. 4) is not recorded separately, nor are Lemmas 1--12, which serve the proofs. The proofs and the manuscript-to-proceedings text were not independently checked.

Results and questions

• Open Problem 1
• Open Problem 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4

Bears on. Problem 2: Open Problem 1 poses the problem's corrected (distinct-moduli) question, and the manuscript answers no part of it. Problem 7: Open Problem 2 poses the problem's question; Theorem 2 shows that for odd $d$ a nontrivial $f$ with $f(x)x^n+d$ reducible for all $n\geq0$ would yield such a covering, without showing that one exists; Theorem 4 constructs a covering by odd moduli greater than $1$ in which a modulus may occur up to three times, so it does not give distinct moduli.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.