Two Questions Concerning Covering Systems

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question_1_4: Asks for the largest guaranteed covering multiplicity attainable while all moduli are distinct and greater than one.

question_1_6: Asks whether one chosen odd modulus may occur twice while every other modulus remains distinct, odd, and nontrivial.

questions_6_1_6_2: Poses a square-free form of Question 1.6 and a square-free minimum modulus question, and asserts that a yes to the first gives a yes to the second, by an argument that has a gap as printed.

section_4_construction: Constructs three covering systems with disjoint modulus sets, whose union covers every integer at least three times with no repeated modulus.

theorem_2: Constructs an (a,b)-primitive three-cover whenever the coprime positive bases do not have power-of-two sum.

theorem_4: Restates Chen's implication from a primitive multiple covering to infinite arithmetic progressions of integers with many prime factors.

theorem_5: For every positive integer b with b+1 not a power of 2, there are infinitely many b-Sierpiński numbers k for which every k b^n+1 has at least three distinct prime divisors.

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Joshua Harrington, Two questions concerning covering systems, International Journal of Number Theory 11 (2015), no. 6, 1739--1750, DOI 10.1142/S179304211550075X. The publisher PDF records receipt on 10 September 2014, acceptance on 23 October 2014, and publication on 2 December 2014; the bibliographic issue year is 2015.

The copy read for this card is the publisher PDF, whose printed pages 1739--1750 are physical PDF pp. 1--12. Its first page prints "© World Scientific Publishing Company".

The introduction reproduces the distinct odd-cover question Problem 7 as Question 1.2 and the minimum-modulus question Problem 2 as Question 1.3. Its comments on their status describe the paper's 2014--2015 setting and are not a fresh status review.

Question 1.4 asks for the largest $N$ such that some covering system with distinct moduli greater than one covers every integer at least $N$ times. Section 4 gives a distinct-modulus three-cover, proving that this parameter is at least three.

Question 1.6 allows one specified odd modulus to repeat at most twice; Section 3 answers the $n=3$ case affirmatively. The later part of the paper introduces $(a,b)$-primitive multiple coverings. Its Theorem 2 constructs a primitive three-cover under an exact power-of-two exception. The paper also restates a consequence attributed there to Chen as Theorem 4, and its Theorem 5 gives, for every base $b$ with $b+1$ not a power of $2$, infinitely many $b$-Sierpiński numbers $k$ with $k\cdot b^n+1$ divisible by three distinct primes for every positive integer $n$. Section 6 poses Questions 6.1 and 6.2, a square-free variant of Question 1.6 and a square-free minimum modulus question for minimum modulus $3$, and asserts an implication between them whose printed argument has a gap, recorded on that page.

Compiled scope

The title page, Questions 1.2--1.6, Sections 3--4 at the level of their stated construction targets, Definitions 5.1 and 5.2, Theorems 2, 4 and 5, Corollary 1, and Questions 6.1 and 6.2 with the argument linking them were read on printed pp. 1739--1749. The exact selected statements are recorded below; read status: claims checked. The long congruence lists, their complete coverage checks, and the proofs were not reconstructed or independently checked, except that the short Section 6 argument was read step by step.

Results and questions

• Question 1.4
• [[covering_systems/harrington_2015_two_questions_covering_systems/question_1_6|Question 1.6 and its $n=3$ case]]
• Section 4 construction
• Theorem 2
• Theorem 4, restated from Chen
• Theorem 5
• Questions 6.1 and 6.2

Bears on.

• Problem 2: the paper reproduces it as Question 1.3, and its Question 1.4 asks for a different parameter, the largest multiplicity of a distinct-modulus covering; the Section 4 construction shows that parameter is at least $3$. Question 6.2 asks for a distinct square-free covering with minimum modulus $3$. The paper proves nothing about the minimum modulus itself.
• Problem 7: the paper reproduces it as Question 1.2. Question 1.6 relaxes it by allowing one odd modulus to occur twice, so an odd covering would answer Question 1.6 yes for every odd $n\geq3$; Section 3 answers Question 1.6 affirmatively for $n=3$, which is not a distinct odd covering. Question 6.1 also allows one odd modulus twice but requires square-free moduli, so it is not a relaxation; an odd covering with square-free moduli would answer both.

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