Necessary condition for odd incongruent coverings II

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block_reduction: Converts distinct-cardinality prime-adic boxes into a controlled family of coordinate blocks and counts their surviving intersections.

forest_union_bound: A forest of pairwise intersections supplies a valid correction to the ordinary union bound, with an explicit nine-edge application.

geometric_obstruction: The forest correction proves the prime-adic box obstruction, with an explicit capacity proof for the source's worst-case assumption.

six_prime_corollary: Proves the required monotonicity and evaluates the limiting obstruction at the five smallest odd primes.

theorem: Transfers the strengthened box obstruction to finite cyclic groups and to distinct covering systems with odd moduli.

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Marc A. Berger, Alexander Felzenbaum and Aviezri S. Fraenkel, Necessary condition for the existence of an incongruent covering system with odd moduli II, Acta Arithmetica 48 (1987), no. 1, 73–79, DOI 10.4064/aa-48-1-73-79. The final page records receipt on 28 June 1985; that is not the publication year.

Source and provenance

The canonical PDF is the published scan obtained through the publisher's free download on 2026-09-05: 231903 bytes. The publisher record labels the download CC BY. The PDF has four physical pages: the first contains printed p. 73, and the next three contain the spreads 74–75, 76–77 and 78–79. All four scans were read visually; no OCR is used for the mathematical transcription. The scan prints no copyright or license line; the publisher's record labels the PDF download "Pobierz zgodnie z CC-BY", which the English site renders "Free download under CC-BY license", naming no version or license URL (https://www.impan.pl/get/doi/10.4064/aa-48-1-73-79, read 2026-10-02).

This is a separate paper from Part I. Its new ingredient is a forest of pairwise intersections that improves the first paper's union bound.

Complete proof chain

The forest lemma gives a valid intersection correction and lists all nine edges used in the source's figure. The block reduction enlarges selected prime-adic boxes, computes the resulting capacities, and proves that the family can be padded to the stated block counts. The geometric proposition then proves the stronger polynomial obstruction. In particular, it justifies the source's assumption that every two-coordinate block meets the remaining product set: under the contradiction hypothesis, the required number of blocks fits inside that set, and replacement preserves coverage.

The main theorem and cyclic-group corollary use Part I's full prime-adic correspondence to transfer the obstruction to integers. Finally, the exponent-free corollary proves the monotonicity claimed in the source and evaluates the worst five-prime case as $1903/960<2$.

These are five complete rewritten proof components, relative to the explicit Part I dependencies and elementary finite counting. The compilation makes the following conventions and implicit arguments explicit: proper boxes and moduli greater than one; $n\ge5$ when the five-coordinate polynomial is evaluated; the separate Part I exclusion for $n<5$; the case $s_1=1$ through a polynomial with no inverse powers; the distinction between arbitrary prime labeling in the geometric proposition and ordered primes in its numerical consequence; and a coordinate-increasing path that preserves the coupled monotonicity constraint. These are compilation-supplied explanations, not a published erratum.

There is also a printed index slip in the recap of Part I's condition. Equation (6) on p. 74 starts its displayed subtraction at $i=2$, whereas the definition $f(x)=\prod_i(1+x_i)-\sum_i x_i$ in equation (2) and Part I's equation (1) require the sum to start at $i=1$. For $(p_1,\ldots,p_5)=(3,5,7,11,13)$, the corrected expression is $1061/495=2+71/495$, exactly the value stated in the next sentence; the expression as printed would instead be $1556/495$. The compilation uses the corrected $i=1$ formula and records this as a source-text correction, not an author-issued erratum.

Relationship and limits

The theorem says that a hypothetical distinct odd covering has a least common multiple divisible by at least six distinct primes. This is a historical necessary condition for Problem 7, not a claimed current best bound and not a resolution of the unrestricted problem.

The later square-free obstruction also converts congruences into geometry, but uses an iterated measure sieve and moment bounds. Its square-free CRT hyperplanes and the prime-power boxes here have different coordinate restrictions. The prime-power extension of that later proof explicitly handles the distinction.

The introductory references to Churchhouse and Porubský are retained as historical attribution; their separate papers are not compiled here. No claim is made about an optimal forest, a sufficient condition for covering, an extension of Part II to all nilpotent groups, or formal verification. Independent mathematical review of this reconstruction is recorded separately from the source's publication.