Is there a covering system all of whose moduli are of the form $p - 1$ for some primes $p \geq 5$ ?

Open — best to date is a honest null, not yet sealed.

number theory · open · formalized (Lean) · 1 attempt

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'difference/covering' (transfer_strength=weak) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'weak'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

If you allow **repeated moduli**, then it’s trivially **yes**: for example [ 0\ (\mathrm{mod}\ 4),\ 1\ (\mathrm{mod}\ 4),\ 2\ (\mathrm{mod}\ 4),\ 3\ (\mathrm{mod}\ 4) ] covers all integers, and (4=5-1) with (p=5).

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 5 11 · open (literature)

theorem erdos_273 : answer(sorry) ↔ ∃ c : StrictCoveringSystem ℤ, ∀ i, ∃ (p : ℕ), p.Prime ∧ 5 ≤ p ∧
    c.moduli i = Ideal.span {↑(p - 1)}

formal-conjectures/273.lean ↗

status

open

evidence

formal

status

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