# P12909: The Vo Sequence Conjecture: A Maximal-Length Property of Half-Gaps Between Consecutive Odd Primes

- ID: `P12909`
- Reference: `vo-sequence-conjecture-maximal-half-prime-gap-sequence`
- Page: https://theoremdb.org/statements/P12909
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/vo-sequence-conjecture-maximal-half-prime-gap-sequence?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let $p_n$ denote the $n$-th odd prime, so $p_1=3,p_2=5,p_3=7,\ldots$. Define $Q(n)=(p_{n+1}-p_n)/2$ for $n\ge1$. For positive integers $x,y,z$, call $(x,y,z)$ a good triple if at least two of $x,y,z$ are equal, or if they are pairwise distinct and $\{x,y,z\}=\{k,k+1,k+2\}$ for some integer $k\ge1$. For $1\le a\le b$, define $Q(a,b)=(Q(a),Q(a+1),\ldots,Q(b))$, with length $b-a+1$. A segment of length at least $3$ is a Vô sequence if every overlapping consecutive triple $(Q(i),Q(i+1),Q(i+2))$, $a\le i\le b-2$, is a good triple. The Vô Sequence Conjecture (VSC) asserts that every Vô sequence has length at most $24$, and that a Vô sequence has length $24$ if and only if $(a,b)=(1,24)$. Equivalently, $Q(1,24)$ is the unique Vô sequence of maximum possible length. Here $Q(1,24)=(1,1,2,1,2,1,2,3,1,3,2,1,2,3,3,1,3,2,1,3,2,3,4,2)$.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `vo-sequence-conjecture-maximal-half-prime-gap-sequence`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.