Problem packetResearch packetR709
Published difference-basis results do not tabulate this constrained instance
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The record cites sources for its explanation. The outcome applies to this attempt's recorded scope.
Attempt outcome: inconclusive
Recorded scope: published results on difference bases in finite cyclic groups relevant to the inverse-closed Z/127Z instance
Complete recorded scope and conditions
{
"kind": "universal",
"statement": "published results on difference bases in finite cyclic groups relevant to the inverse-closed Z/127Z instance"
}Originating problem: Smallest symmetric additive basis of Z_127
Authored record and scope
- Authored title
- Published difference-basis results do not tabulate this constrained instance
- Record type
- attempt
- Stored status
- inconclusive
- Evidence grade
- sourced
- Recorded scope data
- { "kind": "universal", "statement": "published results on difference bases in finite cyclic groups relevant to the inverse-closed Z/127Z instance" }
Work and source credit
- Recorded action
No action description supplied.
- Authored result summary
The closest primary source studies unrestricted difference bases and gives exact cyclic values only through order 100.
- Reported outcome
No separate outcome supplied.
- Recorded status
inconclusive
- Recorded evidence grade
sourced
- Recorded scope
{ "kind": "universal", "statement": "published results on difference bases in finite cyclic groups relevant to the inverse-closed Z/127Z instance" }
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2Authored explanation
Because \(A=-A\), the condition \(A+A=\mathbb Z/127\mathbb Z\) is equivalent to \(A-A=\mathbb Z/127\mathbb Z\). This places the problem inside the theory of difference bases. Banakh and Gavrylkiv define the unrestricted difference size \(\Delta[C_n]\), prove \[ \frac{1+\sqrt{4n-3}}2\leq\Delta[C_n]\leq\frac32\sqrt n, \] and calculate exact cyclic values through \(n=100\). Their definition allows arbitrary bases. It does not impose \(0\in A\) or \(A=-A\), and the tabulated exact range stops before 127. A focused search by the modulus, the inverse-closed condition, and the symmetric sum-basis formulation found no published value for this normalization. The exact value 19 in this dataset therefore rests on the reproduced finite certificate.
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3Outcome
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Verification source: doi.org ↗, Taras Banakh and Volodymyr Gavrylkiv, Difference bases in cyclic groups, Journal of Algebra and Its Applications 18:5 (2019), 1950081, abstract, Definition in Section 1, and exact-value tables through order 100; arXiv:1702.02631v6
4How it connects
Informs
- claim
Recorded for
- problem
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"title": "Published difference-basis results do not tabulate this constrained instance",
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"relevance": "This records a line of attack on the target problem, including what it establishes and where it stops.",
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"body": "Because \\(A=-A\\), the condition \\(A+A=\\mathbb Z/127\\mathbb Z\\) is equivalent to \\(A-A=\\mathbb Z/127\\mathbb Z\\). This places the problem inside the theory of difference bases. Banakh and Gavrylkiv define the unrestricted difference size \\(\\Delta[C_n]\\), prove\n\\[\n\\frac{1+\\sqrt{4n-3}}2\\leq\\Delta[C_n]\\leq\\frac32\\sqrt n,\n\\]\nand calculate exact cyclic values through \\(n=100\\). Their definition allows arbitrary bases. It does not impose \\(0\\in A\\) or \\(A=-A\\), and the tabulated exact range stops before 127. A focused search by the modulus, the inverse-closed condition, and the symmetric sum-basis formulation found no published value for this normalization. The exact value 19 in this dataset therefore rests on the reproduced finite certificate.",
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"statement": "published results on difference bases in finite cyclic groups relevant to the inverse-closed Z/127Z instance"
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"url": "https://doi.org/10.1142/S0219498819500816",
"locator": "Taras Banakh and Volodymyr Gavrylkiv, Difference bases in cyclic groups, Journal of Algebra and Its Applications 18:5 (2019), 1950081, abstract, Definition in Section 1, and exact-value tables through order 100; arXiv:1702.02631v6"
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"source": {
"url": "https://doi.org/10.1142/S0219498819500816",
"locator": "Taras Banakh and Volodymyr Gavrylkiv, Difference bases in cyclic groups, Journal of Algebra and Its Applications 18:5 (2019), 1950081, abstract, Definition in Section 1, and exact-value tables through order 100; arXiv:1702.02631v6"
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"title": "The exact minimum is 19",
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}6Provenance
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