Sun 2007 covering numbers

../

conjecture_1_1: Sun's conjecture, the converse of his Theorem 1.1 for primitive covering numbers: each one can be written as p_1^{a_1} ... p_r^{a_r} with distinct primes so that the product of (a_t + 1) over t < s is at least p_s - [r != s] for every s; the paper notes it is stronger than the Erdős-Selfridge conjecture.

corollary_1_2: Sun's corollary that for every r = 2, 3, ... there are infinitely many primitive covering numbers with exactly r distinct prime divisors, deduced from his Theorem 1.3 and Dirichlet's theorem.

corollary_1_3: Sun's answer to Erdős's 1980 question: there are infinitely many n, namely n = 2^{p-1}p for the odd primes p, such that among the subsets of the divisors of n greater than one only the whole set can be the moduli of a cover of the integers with distinct moduli.

theorem_1_1: Sun's sufficient condition for a covering number: for distinct primes p_1, ..., p_r and positive exponents a_1, ..., a_r, if the product of (a_t + 1) over t < s is at least p_s - [r != s] for every s, then p_1^{a_1} ... p_r^{a_r} is a covering number.

theorem_1_2: Sun's characterization of exponent tuples: for positive a_1, ..., a_r there are primes p_1 < ... < p_r making p_1^{a_1} ... p_r^{a_r} a covering number exactly when r = 2 and a_1 >= 2, or r = 3 and max(a_1, a_2) >= 2, or r >= 4.

theorem_1_3: Sun's construction of primitive covering numbers: for primes 2 = p_1 < ... < p_r, r > 1, with p_t - 1 dividing p_{t+1} - 1 for 0 < t < r - 1 and p_r >= (p_{r-1} - 2)(p_{r-1} - 3), an explicit product of powers of these primes, ending in p_r to the first power, is a primitive covering number.

theorem_1_4: Sun's classifications: an integer n > 1 with at most two distinct prime divisors is a primitive covering number exactly when n = 2^{p-1}p for an odd prime p; a multiple of 3 with exactly three is one exactly when n = 2 * 3^{(p-1)/2} p for a prime p > 3; and four further explicit families are primitive.

---

Zhi-Wei Sun, On covering numbers, Integers 7 (2007), no. 2, A33, also printed in Combinatorial Number Theory (de Gruyter, Berlin, 2007), 443--453, DOI. The copy read for this card is the 11-page arXiv preprint math/0601017v2 (9 September 2006), headed as to appear in INTEGERS; its theorem numbering and pages are cited below. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0601017), every other right reserved.

An integer n is a covering number (Definition 1.1, p. 2) if the integers can be covered by residue classes whose moduli are distinct divisors of n greater than one, and primitive (Definition 1.2, p. 4) if no proper divisor of n is a covering number. Theorem 1.1 (p. 3) gives a sufficient condition for p_1^{a_1}...p_r^{a_r} to be a covering number, namely prod_{0<t<s}(a_t+1) >= p_s - [r != s] for each s, where [P] is 1 or 0 as P holds or not (condition (1.3)); Theorem 1.2 (p. 4) characterizes exactly which exponent tuples (a_1,...,a_r) occur for some increasing primes p_1 < ... < p_r (r = 2 with a_1 >= 2, r = 3 with max{a_1,a_2} >= 2, or r >= 4). Theorem 1.3 (p. 4) constructs primitive covering numbers from primes 2 = p_1 < ... < p_r, r > 1, with p_t - 1 | p_{t+1} - 1 for 0 < t < r - 1 and p_r >= (p_{r-1}-2)(p_{r-1}-3), and Corollary 1.2 (p. 4) deduces via Dirichlet's theorem that for every r >= 2 there are infinitely many primitive covering numbers with exactly r distinct prime factors. Theorem 1.4 (p. 4) classifies the primitive covering numbers with at most two prime divisors as exactly 2^{p-1}p for odd primes p, and those divisible by 3 with exactly three prime divisors as exactly 2*3^{(p-1)/2}p for primes p > 3, and supplies four further explicit families. Corollary 1.3 (p. 5) answers Erdős's 1980 question affirmatively: there are infinitely many n for which the only subset of D_n = {d >= 2 : d | n} that can serve as the full set of moduli of a distinct-moduli cover of Z is D_n itself, proved for n = 2^{p-1}p from Theorem 1.4 (i) together with Simpson's proof of Znám's conjecture. The paper also conjectures (Conjecture 1.1, p. 5) that every primitive covering number can be written as p_1^{a_1}...p_r^{a_r}, with distinct primes p_t, so that condition (1.3) holds, and notes (Remark 1.4) that this is stronger than the Erdős--Selfridge conjecture.

Read status: claims checked for Theorems 1.1 to 1.4, Corollaries 1.2 and 1.3 and Conjecture 1.1, read clause by clause on the page images of the print, with the proofs followed; Simpson's theorem, Dirichlet's theorem and the earlier covering results the paper cites are not proved in it. Nothing here is independently reviewed.

Source: https://arxiv.org/abs/math/0601017.

Bears on. #1189: Corollary 1.3 (p. 5) and its proof show that for each odd prime p the divisors of 2^{p-1}p greater than one form a covering set no proper subset of which is a covering set, answering the problem's last question yes; the paper does not address the problem's other questions. #7: Conjecture 1.1 (p. 5) would, by the paper's Remark 1.4, imply the Erdős--Selfridge conjecture that no cover with distinct moduli greater than one has all moduli odd, a negative answer to the problem; the paper proves neither.

Results.

• Theorem 1.1 (p. 3): condition (1.3) makes p_1^{a_1}...p_r^{a_r} a covering number; with Remark 1.1 and Corollary 1.1 (p. 3).
• Theorem 1.2 (p. 4): the exponent tuples of covering numbers with increasing primes.
• Theorem 1.3 (p. 4): primitive covering numbers from prime chains; with Remark 1.2.
• Corollary 1.2 (p. 4): infinitely many primitive covering numbers with exactly r prime divisors, for each r >= 2.
• Theorem 1.4 (p. 4): the classifications with at most two prime divisors, and with three for multiples of 3, and further families; with Remark 1.3.
• Corollary 1.3 (p. 5): the answer to Erdős's 1980 question.
• Conjecture 1.1 (p. 5): the converse of Theorem 1.1 for primitive covering numbers.

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