Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
In the sets have , below the Cauchy–Davenport bound
Statement refuted
The Cauchy-Davenport lower bound can fail for a composite modulus.
Facts & Assumptions
Given: the modulus and the set .
For prime and nonempty , Cauchy--Davenport gives (Cauchy–Davenport: for prime and nonempty , ).
Counterexample
The four sums are , , and , so and therefore .
The Cauchy-Davenport lower bound would be , so the inequality fails for this composite modulus.
The failure occurs outside the theorem's prime-modulus hypothesis: here the modulus is , not a prime , so [L1] does not apply.
Depends on
- Cauchy–Davenport: for $p$ prime and nonempty $A,B\subseteq\mathbb{Z}/p$, $\lvert A+B\rvert\ge\min\{p,\lvert A\rvert+\lvert B\rvert-1\}$
- The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
- For every natural $n$, $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold
- For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field
- Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- N. Alon, Combinatorial Nullstellensatz, Theorem 3.2 (standard reference, not scraped)