Let $k \geq 3$ . Is there a choice of congruence classes $a_{p} (mod p)$ for every prime $p$ such that all sufficiently large integers can be written as $a_{p} + tp$ for some prime $p$ and integer $t \geq k$ ?

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

machinery: covering-system,prime-distribution,large-prime-gaps-FGKT,consecutive-integer-window,probabilistic-Borel-Cantelli-patching,sieve/Brun-Titchmarsh,S-unit-residue-cover

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

As of **January 16, 2026**, this is **open for every fixed (k\ge 3)** [[nomath]](and “even the case $k=3$ seems difficult”)[[/nomath]]. It appears as an Erdős–Graham problem (Erdős Problem #279 in the community database). ([Erdős Problems][1])

candidate solution ↗

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_279 :  answer(sorry) ↔ ∀ k : Nat, k ≥ 3 →
    ∃ a : Nat → Nat, ∃ N : Nat, (∀ p : Nat, p.Prime → a p < p) ∧
    ∀ n ≥ N, ∃ p : Nat, ∃ t ≥ k, p.Prime ∧ n = a p + t * p

formal-conjectures/279.lean ↗

status

open

evidence

formal

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