Nielsen 2009 covering system smallest modulus 40

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arrow_finitization: Replaces the mnemonic infinite prime-tree recursion by a finite, collision-safe family of residue classes.

construction_ledger: Audits the reconstructed residual holes and isolates the unresolved prime-37 allocation before the arrow tails are made finite.

initial_primes_2_7: Removes every modulus below 40 from the initial tree and records the exact packages and residual holes created by the first four primes.

later_signature_certificate: Gives deterministic input blocks and conditional modulus-injectivity checks for the prime-17 and prime-29 through prime-103 construction stages.

main_theorem: Nielsen's unnumbered main result, stated in the abstract, that there is a finite covering of the integers by congruence classes with distinct moduli greater than one whose smallest modulus is 40.

notation: Gives the exact residue-class and modulus semantics of Nielsen's nested prime-tree expressions, separately from the sets they are used to cover.

prime_11_template: Gives the ten exact input packages that fill Nielsen's 11-arrow on the selected halves of the modulus-6 and modulus-18 holes.

prime_13_template: Reflects the prime-11 construction across the other odd class modulo 4 and supplies the final two inputs by modified 11-arrow packages.

prime_17_template: Gives Nielsen's sixteen prime-17 input packages and records the two deleted low-modulus branches and the one residual partial branch.

prime_19_template: Gives the seventeen filled inputs and selected-input tail used at prime 19, including the exact nested 11- and 13-arrow packages.

prime_23_template: Gives the twenty-two packages that fill the modulus-24 hole and isolates the exact Nielsen template reused by Owens.

primes_29_37: Completes the prime-29 and prime-31 templates and records the unresolved input allocation in the source's prime-37 template.

primes_41_67: Reconstructs the local prime-41 through prime-67 schedules, with the prime-59 branch conditional on the unresolved prime-37 input pool.

primes_71_103: Reconstructs the prime-71 through prime-103 schedules, with the prime-89 branch conditional on the unresolved prime-37 and prime-59 interface.

template_signature_certificate: Partitions the unbounded prime-exponent regions of Nielsen's reusable templates and proves that their regular moduli are pairwise distinct.

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Pace P. Nielsen, A covering system whose smallest modulus is 40, Journal of Number Theory 129 (2009), no. 3, 640–666, DOI 10.1016/j.jnt.2008.09.016.

The copy read for this card is the complete 25-page author version. Nielsen's publication page identifies this as the preprint version and separately gives the journal citation and DOI; the journal issue is that of March 2009. Result pages cite physical pages of this author PDF, not the journal's pages 640–666. No claim of byte-equivalence to the published version is made. The author's version prints no copyright or license line, and the author's publication page that labels it the preprint states no terms; the term is unstated.

Published result and method

The paper states and describes a finite covering of every integer by residue classes whose moduli are all different and whose least modulus is exactly $40$. It organizes the construction as rooted prime-power trees. An entry at a leaf represents a residue class, multiplication records intersection of compatible prime-power conditions, and an upward arrow records a finite repeated descent through one prime tree. The source estimates (p. 24) that, with $p=107$ closing the arrows, the cover has many more than $(p-1)^{25}>10^{50}$ classes; that size estimate is not derived from the partial reconstruction compiled here.

The source and its reconstruction are organized in the following order.

• Prime-tree notation gives the exact CRT semantics.
• Arrow finitization separates the regular finite descent from its terminal closure and proves a collision-safe realization.
• The primes 2, 3, 5, and 7 creates and records the initial holes.
• The exact ordered templates for 11, 13, 17, 19, and 23 are retained separately because later constructions reuse them.
• The regular-signature certificate expands those reusable packages into exact unbounded prime-exponent regions.
• Primes 29 through 37 proves the prime-$29$ and prime-$31$ templates and identifies the missing distinct-modulus allocation in the source's second prime-$13$ package at prime $37$.
• Primes 41 through 67 and 71 through 103 reconstruct their local schedules; the prime-$59$ and prime-$89$ branches are conditional on the missing prime-$37$ pool.
• The later regular-signature certificate fixes every ordered block and checks the exceptional unbounded exponent regions.
• The construction ledger records the package-count and no-duplicate invariants at their proved or conditional scopes.
• The main-result page states the published theorem (abstract, p. 1) and points to the sections of the paper that construct it.

Nielsen remarks on physical p. 23 that the single prime $107$ can close every arrow; on physical p. 7 he says a single fixed prime suffices but does not prove it. The arrow lemma compiled here uses the simpler alternative allowed on physical p. 7: a fresh terminal prime for each realized arrow occurrence. That settles finiteness and terminal collisions once the regular symbolic construction is valid; it cannot repair a collision among regular modulus patterns.

Reconstruction boundary

On physical p. 20 the source says that a retained partial $5^\uparrow$ package can be used to fill a second $13^\uparrow$, but it does not give the inputs. The direct ordered completion fails the distinct-modulus check when packages already containing prime-$5$ arrows are put under another prime-$5$ input. No replacement allocation was reconstructed here. The prime-$37$ page gives the exact obstruction and propagates it only to the dependent stages. This records a limitation of this compilation, not an author erratum and not a claim against the published theorem.

Owens's 2014 thesis reports a construction with least modulus $42$ in the same notation; see its card.

Bears on.

• Problem 2: the paper's main result (unnumbered; abstract, p. 1) exhibits a finite covering system with distinct moduli greater than $1$ whose least modulus is $40$. If the least modulus of such systems is bounded, the bound is therefore at least $40$. It does not answer whether the least modulus can be arbitrarily large; the paper (p. 1) calls that question open and says its method leads the author to believe the answer is negative.

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