A positive odd integer $m$ such that none of $2^{k} m + 1$ are prime for $k \geq 0$ is called a Sierpinski number. We say that a set of primes $P$ is a covering set for $m$ if every $2^{k} m + 1$ is divisible by some $p \in P$ .Are there Sierpinski numbers with no finite covering set of primes?

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

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

machinery: covering-system,Sierpinski-Riesel-number,algebraic-factorization-x4plus1-Aurifeuillian,Fermat-prime-distribution,perfect-power,non-covering-compositeness-certificate,multiplicative-order-mod-p

use this record

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

As of January 2026, **no example is known that is *proved* to be a Sierpiński number while also being *proved* to admit *no* finite prime covering set**. In other words: **the existence of “non‑covering” Sierpiński numbers is an open problem.** ([Erdős Problems][1])

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 11 · solved (literature)

theorem erdos_1113.variants.infinitely_many_sierpinski :
    Set.Infinite {k : ℕ | k.IsSierpinskiNumber}

formal-conjectures/1113.lean ↗

oeis

A076336 — (Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite.78557,271129,271577,322523,327739,482719,575041,603713,903983,934909,965431,1259779,1290677,1518781,1624097,1639459,1777

links

Sierpinski number · reference

status