ethosnode

ALT Conjectures announcement on an ivory background with green and black type. Headline: Erdős Problem 1032 — DISPROVED. Open for over 50 years. Badge: Verified in Lean. For edge-critical four-chromatic graphs on n vertices, the minimum degree δ(G) is o(n): for every ε > 0, δ(G) < εn for all sufficiently large n. Two diagrams illustrate edge-criticality: K₄ requires four colours; deleting an edge allows a three-colouring, with the two now nonadjacent vertices sharing a colour. K₄ is a small illustration, not the asymptotic proof. Credit: EthanYang. conjectures.io.

ALT Headline: Wilf’s inequality — CASE PROVED. Conductor at most four times multiplicity. Conjectures announcement for Wilf’s inequality: c≤4m implies eL≥c for proper numerical semigroups. Definitions: m is the smallest positive element, c the first integer after the final gap, e the number of minimal generators, and L the number of elements below c, including zero. Two rows show membership in the semigroup generated by 5 and 6 from zero through 29, using solid green dots for members and hollow gray dots for gaps. The final gap is 19, with every integer from 20 onward present. The example has m=5,e=2,L=10, so eL=20=c=4m. Formally verified in Lean. Credit: Elias. Historical caption: 48-YEAR-OLD CONJECTURE · SINCE 1978.

ALT Headline: Erdős Problem 1057 — PROVED. Conjectures announcement for Erdős 1057. The counting theorem is C(x)=x^(1-o(1)), where C(x) counts Carmichael numbers at most x. For each epsilon>0, eventually C(x)≥x^(1-epsilon). A factorization diagram connects 561 to 3,11,17. Their predecessors 2,10,16 divide 560, and all 320 coprime residue classes satisfy a^560≡1 modulo 561. Ivory background, bold serif headline, large green verdict and an outlined Verified in Lean badge. Formally verified in Lean. Credits: JenW1N, Liam Kruer and gus.