On the Range of a Covering Function

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corollary_1_1: Shows that every modulus in a constant covering function divides another modulus, forcing equality of the two largest ordered moduli.

corollary_1_2: Excludes every nontrivial residue class from the range when the moduli maximal under divisibility are distinct.

theorem_1_1: Forces a modulus n_t to divide another modulus when the covering-function range lies in one residue class mod m and m n_t does not divide the least common multiple of the moduli.

theorem_1_2: Identifies two distinct-modulus residue systems whose covering functions are congruent modulo an integer not dividing the least common multiple of all their moduli.

theorem_1_3: Shows that for a weighted covering function whose least period modulo m is not divisible by d, either m divides an explicit weighted sum over the moduli divisible by d, or the residues a_s mod d of those classes take at least p(d) values, p(d) the least prime factor of d.

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Zhi-Wei Sun, On the range of a covering function, Journal of Number Theory 111 (2005), no. 1, 190--196, DOI 10.1016/j.jnt.2004.11.004.

The copy read for this card is the seven-page arXiv:math/0409279v2, identified on its first page as the final version of 21 September 2004 for the Journal of Number Theory. The result pages below use that version's PDF pagination. The publisher record establishes the 2005 journal identity; the publisher PDF was not compared with that v2. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0409279), every other right reserved.

For a finite system $\{a_s(n_s)\}_{s=1}^k$ with $k>1$ and positive integer moduli $n_s$, where $a(n)$ denotes the residue class $a\pmod n$, the covering function is

$$w(x)=\left|\{1\leq s\leq k:x\in a_s(n_s)\}\right|.$$

Theorem 1.1 forces a divisibility relation whenever the range of $w$ lies in one residue class modulo an integer $m$. Its two immediate corollaries treat constant covering functions and systems whose divisibility-maximal moduli are distinct. Theorem 1.2 gives a modular uniqueness criterion for two systems with distinct moduli, and Theorem 1.3 refines Theorem 1.1 to integer-weighted covering functions taken modulo an integer $m$.

When all moduli are distinct, the divisibility-maximal moduli are automatically distinct, so Corollary 1.2 shows that the covering multiplicity $w(x)$ takes values of both parities. This is context for Problem 7, not an implication about its modulus condition: $w(x)$ counts how many congruences contain $x$, whereas Problem 7 asks whether the moduli themselves can all be odd. A hypothetical distinct all-odd-modulus cover may still have covering multiplicities of both parities.

Compiled scope

The definitions, Theorems 1.1--1.3, Corollaries 1.1--1.2 and Remarks 1.1--1.4 were read on PDF pp. 1--4. The theorem and corollary statements are restated on the result pages below. Section 2, PDF pp. 4--6, contains the proofs; they were not reconstructed or independently checked here.

Results

• Theorem 1.1
• Corollary 1.1
• Corollary 1.2
• Theorem 1.2
• Theorem 1.3

Bears on. Problem 7: context only. Corollary 1.2, with Remark 1.2 (PDF p. 3), shows that a cover with distinct moduli does not cover every integer an odd number of times. It says nothing about whether the moduli of such a cover can all be odd.

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