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CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26
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A Steiner triple system can exist only when v≡1 or 3(mod6)

Statement

If a Steiner triple system of order v exists, then v≡1 or 3(mod6).

Facts & Assumptions

Given: A Steiner triple system of order v.

[L1]

A Steiner triple system is a 2-(v,3,1) design (Steiner triple systems).

[L2]

For a 2-design, the numbers r=λ(v−1)/(k−1) and b=vλ(v−1)/(k(k−1)) are integers (The standard divisibility conditions for a 2-design).

Proof

technique · direct
1.1

Substituting k=3 and λ=1 into [L2] gives r=(v−1)/2∈N, so v is odd.

L1L2algebra
1.2

The same substitution gives b=v(v−1)/6∈N, so 3 divides v(v−1)/2.

L1L2algebra
2.1

Among the odd residue classes modulo 6, only 1 and 3 make v(v−1)/2 divisible by 3. Hence v≡1 or 3(mod6).

step 1.1step 1.2algebra∎

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Dependency tree · two levels

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Sources