TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26‡ rests on unproved material

How 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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

A Steiner triple system exists exactly for orders $v > 3$ with $v \equiv 1$ or $3 (m o d 6)$

Statement

A Steiner triple system of order $v$ exists if and only if $v > 3$ and $v \equiv 1$ or $3 (m o d 6)$ .

Facts & Assumptions

Given: A natural number $v$ .

[L1]

If a Steiner triple system of order $v$ exists, then $v \equiv 1$ or $3 (m o d 6)$ (A Steiner triple system can exist only when $v \equiv 1$ or $3 (m o d 6)$ ).

[L2]

For every natural number $m \geq 1$ , Bose's construction yields a Steiner triple system of order $6 m + 3$ (Bose's construction gives a Steiner triple system of order $6 m + 3$ for $m \geq 1$ ).

‡ [L3]

For every integer $m \geq 1$ , Skolem's construction yields a Steiner triple system of order $6 m + 1$ (Skolem's construction gives a Steiner triple system of order $6 m + 1$ ‡).

[L4]

A Steiner triple system of order $v$ is a $2$ - $(v, 3, 1)$ design, and a $2$ -design requires $2 \leq 3 < v$ (Steiner triple systems, A $2$ - $(v, k, λ)$ design).

Proof

technique · direct

1.1

If a Steiner triple system of order $v$ exists, then [L4] gives $v > 3$ , and [L1] gives $v \equiv 1$ or $3 (m o d 6)$ .

L1L4

1.2

Conversely, suppose $v > 3$ and $v \equiv 3 (m o d 6)$ . Then $v = 6 m + 3$ for some natural number $m$ , and $v > 3$ forces $m \geq 1$ . So [L2] gives a Steiner triple system of order $v$ .

L2algebra

1.3

Suppose instead that $v > 3$ and $v \equiv 1 (m o d 6)$ . Then $v = 6 m + 1$ for some integer $m \geq 1$ , because $v \equiv 1 (m o d 6)$ and $v > 3$ force $v \geq 7$ . Now [L3] gives a Steiner triple system of order $v$ .

‡ L3algebra

2.1

The two congruence classes $1$ and $3$ modulo $6$ exhaust the converse assumption, so steps 1.2 and 1.3 prove that direction. Together with step 1.1, this proves the theorem.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Jonathan Davidson, Steiner Triple Systems (standard reference, not scraped)

A Steiner triple system exists exactly for orders v>3v>3 with v≡1v≡1 or 3(mod6)3(mod6)