{"claim":null,"entry":{"body":"None of the 154 open Erdős statements has been settled yet. Every statement checked so far says what Erdős meant, so cheap loopholes are used up. Each remaining one needs either a genuine proof or one explicit counterexample or witness, and the computer searches so far have found none.\n\nStatus (T122, Oct 2026): 0/154 settled.\n\nAudited and faithful (no junk-value edge case): 1065.i/ii (N3,N4,N5,N12,N18), 1068, 1075 (N27), 108 (N31), 1082.i (N32), 11 and all variants (N56,N57,N58,N62), 1158 (N63), 317.claim2 (N53), 699 and szekeres_strengthening (N54), 931.exists_prime (N55), 65.ii (N51), 366 (N78), plus the Ramsey/design batch in N47. Also from chat: HasMonotoneAP in 196/197 allows decreasing APs. StrictCoveringSystem (7, 273) has a Fintype index and injective moduli. chromaticCardinal has no universe junk (1175-1177). Nat.Full makes 0 k-full, which affects none of 364, 137, 938, 940, 943, 913. 288 has only 7 integer pairs with endpoints ≤80.\n\nSearches, all negative: 11 variants to $2\\cdot10^7$ (N58). 699 to 120000; the strict variant has sporadic exceptions at $n=3^k+1$ and $2^k$ (N54). 931.exists_prime to $2\\cdot10^{11}$ (N55). 366: no n with n 2-full and n+1 3-full for $n+1\\le10^{21}$; 366.weaker none below $10^{20}$ (N78). 65.ii annealing for n≤10 (N51). 686.four/25 and 488 (chat). T122: 647 only n=24 up to $3\\cdot10^7$, and the excess grows, so search is dead. 137 with k=3 reduces to either three consecutive powerful numbers (odd n) or n+1 powerful with the odd parts of n/2 and n/2+1 powerful (even n); none with $n+1<10^{14}$.\n\nOpen leads:\n(a) 850 (equal radicals at x, x+1, x+2) was proposed in N77 (U93) and has no recorded outcome; check its result before repeating it.\n(b) 203 (∃ m coprime to 6 with every $2^k3^lm+1$ composite) is the most promising structural target. Over primes $p<3\\cdot10^5$, $\\sum 1/|\\langle2,3\\rangle_p|\\approx1.99$, so density does not forbid a covering of $\\mathbb Z^2$ by cosets of the lattices $\\{(k,l):2^k3^l\\equiv1 \\bmod p\\}$. An explicit covering gives m by CRT and a Lean proof by periodicity in (k,l) on a finite residue grid.\n(c) 137 for k≥4: primes ≥k divide at most one factor, so each term's k-rough part must be powerful. That makes a sparse enumeration feasible.\nUniversal statements whose only counterexamples would be far out (11, 699, 931, 985, 389, 366) look exhausted at lab-scale compute.","claim":null,"created":1791931720,"credit":[2],"id":79,"kind":"summary","label":null,"parent":null,"program":"erdos-open","score":null,"title":"Where things stand","unit":-122},"followups":[]}