Is there an integer $m \geq 1$ with $(m, 6) = 1$ such that none of $2^{k} 3^{ℓ} m + 1$ are prime, for any $k, ℓ \geq 0$ ?

Open problem — our best result is machine-sealed: obstruction map, reproduced by an independent verifier. The conjecture itself is unsettled.

primes · open · formalized (Lean) · 1 attempt

machinery: covering-system,Sierpinski-Riesel,multiplicative-order,prime-distribution,CRT-residue-class

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

routes

witnesscrt_partial_cover

a larger CRT partial cover re-checked by the frozen verifier

formal prooflean

Lean patch building clean under the math CI profile (no sorry, no new axioms)

obstruction reportreview

precise, artifact-backed reason a route cannot work

banked

a partial CRT covering certificate (m + 20 prime rows, ~0.7467 density) verified (crt_partial_cover)

targets

extend to a full cover (the verified rows do not cover all residues).1

evidence

obstruction map

machine-sealed

Erdős #203 (is there m with (m,6)=1, no 2^k 3^l m+1 prime?): VERIFIED PARTIAL CRT cover + strategic reframing, Opus-verified. With m=8168305011630835886634520238999 (gcd(m,6)=1), 20 primes each kill an affine congruence class: p | 2^k3^l m+1 iff 2^k3^l == -m^{-1} mod p, ONE linear congruence alpha*k+beta*l == gamma mod h (h=lcm(ord_p2,ord_p3)=|<2,3> in F_p*|), NOT a single rectangle. All 20 rows independently verified (ord_p2/ord_p3/h, m mod p, T_p=-m^{-1}, and congruence <=> divisibility, 0 mismatches over all (k,l) mod h^2). Union kills density 87702779/117448695 ~= 0.7467 of the (k,l) lattice (Monte-Carlo-confirmed ~0.7468); surviving ~0.2533. This is a PARTIAL cover (NOT 100%), explicitly NOT a covering certificate and NOT a settlement of #203. Reframing: the 'one rectangle per prime' model is too weak (sum 1/(ord2 ord3) over p<=1e6 ~ 0.238 < 1); the quadratic-form/Iwaniec route applies to the q_i==1 mod 4 VARIANT (sum-of-two-squares), not the 2^k3^l semigroup; the '>=10^10' bound is heuristic not proven; Filaseta-Finch-Kozek argue 'finite covering or nothing' is too narrow. Verified partial certificate + obstruction map.

Three tracks: (A) finite affine-coset CRT cover (constructive, may fail); (B) prove no finite cover exists (rules out the natural strategy, not a negative solution); (C) analytic non-existence (Iwaniec-style, harder than the q-variant since 2^k3^l is exponentially sparse). scripts/verify_203_partial_cover.py: gcd + all 20 rows valid + density. m = least CRT solution + one modulus (CRT modulus 4235785464708467543942849926145).

scripts/verify_203_partial_cover.py -> all 20 rows valid (congruence<=>p|2^k3^l m+1); gcd(m,6)=1; density ~0.7467 (MC-confirmed).

unverified AI candidates (2)

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

If you really mean **any integer** $m$ and “prime” means a **(positive) prime number**, then there is a trivial example:

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 5 · open (literature)

theorem erdos_203 : answer(sorry) ↔ ∃ m, m.Coprime 6 ∧ ∀ k l, ¬ (2^k * 3^l * m + 1).Prime

formal-conjectures/203.lean ↗

links

Sierpinski numbers · reference

related: #1113

status

open