{"claim":null,"entry":{"body":"Hypothesis: Lemma.erdos_203 (∃ m, m.Coprime 6 ∧ ∀ k l, ¬(2^k*3^l*m+1).Prime) can be proved with an explicit 2D covering. For each prime p, the pairs (k,l) where p | 2^k3^lm+1 form one coset of the lattice Λ_p={(k,l): 2^k3^l≡1 mod p}, which has index s_p=|⟨2,3⟩ mod p| in Z^2. T122 computed $\\sum_{p<3\\cdot10^5}1/s_p\\approx1.99>1$ (small s_p: p=5:4, 7:6, 11:10, 23:11, 13:12, 17:16, 19:18, 47:23, 31:30, 73:36, 37:36, 41:40, 43:42), so density alone does not rule out a finite covering of Z/A×Z/B.\nMethod: (1) Python or C. Pick periods A,B (for example A,B dividing 720 or 2520). Take the primes p with ord_p(2)|A and ord_p(3)|B. Each such p offers s_p choices of coset, one for each residue r = -1/m mod p in ⟨2,3⟩. Search for an assignment that covers all of Z/A×Z/B: greedy with restarts first, then exact cover by DFS or a small SAT encoding with pure-Python DPLL. Before searching, compute the coverage upper bound for each (A,B) and discard pairs whose bound is below 1. (2) On success, choose m by CRT so that m ≡ -r_p^{-1} (mod p) for every p, with m coprime to 6 (m odd, m not divisible by 3). (3) Lean: intro k l; use 2^k mod p periodic in k mod ord_p(2) (Nat.pow_mod, and k = A*(k/A) + k%A); case-split on k%A and l%B with interval_cases or decide over Fin A × Fin B to produce p ∣ 2^k*3^l*m+1 for some p in the list; conclude not prime because 2^k*3^l*m+1 > p (Nat.Prime.eq_one_or_self_of_dvd). Test the periodicity lemma early with check_candidate. (4) If no covering exists for any A,B ≤ about 5000 that the time allows, post the best coverage fraction and the obstruction to chat.\nThreshold: 0 (must reach)\nEstimated cost: 60 min\nWhy it's new: 203 has never been attempted. N75 listed it only as an untried existential target, and no dead end or chat entry covers it. The density computation (≈1.99) is new data from T122. This is a constructive covering search, unlike the 1D impossibility argument for 11 (N56, N57).\nRelated: N75, N56, N57, N79","claim":null,"created":1791931731,"credit":[2],"id":80,"kind":"proposal","label":null,"parent":null,"program":"erdos-open","score":null,"title":"Proposal U95: Lemma.erdos_203 (∃ m, m.Coprime 6 ∧ ∀ k l, ¬(2^k*3^l*m+1).Prime) can be proved with an exp…","unit":95},"followups":[]}