{"schema":"vela.problem-packet.v0.1","problem":366,"statement":"Are there any $2$-full $n$ such that $n+1$ is $3$-full? That is, if $p\\mid n$ then $p^2\\mid n$ and if $p\\mid n+1$ then $p^3\\mid n+1$.","status":"open","seam":"sealed","closureRoutes":[{"type":"verified_computation","verifierKind":"python:verify_366_mordell.py","note":"Mordell-curve reduction + Pell witness re-check (pinned tier: scripts/requirements-verifiers.txt)"},{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"}],"obligations":[{"findingId":"vf_cd5fe218ed92cc31","banked":"the unconditional Mordell-curve reduction + Pell correction + the two analytic walls verified","open":"the analytic walls block an unconditional bound; closing them is open.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_9c20152754234526","kind":"reduction","claim":"Erdős #366 (is there a 2-full n with n+1 3-full?): UNCONDITIONAL Mordell-curve reduction + effective-abc finite reduction, Opus-verified. Normal forms: 2-full n=q^3 y^2 (q squarefree), 3-full n+1=d z^3 (d cubefree, rad(d)|z), so #366 is dz^3-q^3y^2=1. Setting X=dqz, Y=dq^3y gives the exact equivalence to integral points on the Mordell curve Y^2=X^3-d^2q^3 with divisibility filters dq|X, dq^3|Y, rad(d)|X/(dq) (Mordell identity verified symbolically). Under EXPLICIT abc (c<=K_e rad(abc)^{1+e}): rad(n)<=n^{1/2}, rad(n+1)<=(n+1)^{1/3} => n+1 <= B_e=K_e^{6/(1-5e)}, and rad(d)|z forces d<=m^{2/5}, so #366 becomes a FINITE explicitly-bounded search over q<=B^{1/3} (squarefree), d<=B^{2/5} (cubefree) genus-1 curves with explicit height bounds. Unconditional: the normal form, Mordell equivalence, filters, heights. Conditional: a NUMERICAL B (ordinary abc gives finiteness in principle; explicit abc gives a computable search). Pell correction: x^2-8y^2=1 gives a SQUARE on top (8y^2 not generally 3-full, e.g. 8*35^2=9800=2^3*5^2*7^2), the wrong order for #366 — the asymmetry is geometric (genus-1 Mordell, not genus-0 Pell). Conditional finite reduction; no witness found, not a proof. Credits abc-finiteness comment + Aktas-Murty + Hall/Elkies.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_4ae27dd8d6030e13","kind":"reduction","claim":"Erdős #366: the effective-finiteness route is BLOCKED at a named kernel bound, Opus-verified. For bounded kernel D=d^2 q^3 <= K, Baker's method on Y^2=X^3-D gives an explicit (astronomical but effective) search bound n+1 <= ceil(exp(3*(1e7*sqrt3*K^{1/3})^{1e7})); enumerate D<=K (v_p(D) in {0,2,3,4}), recover (d,q), compute the filtered Mordell points. But there is NO unconditional bound on D: the normal form gives only a LOWER bound n+1>=D^{5/9} (since D<=(n+1)^{9/5}), not an upper bound on D; fixed-curve Baker/elliptic-Chabauty bound points per curve but not across the MOVING family (generic moving Mordell has unbounded D, e.g. (t^2)^3-(t^3-1)^2=2t^3-1); Thue-Mahler needs fixed prime support; unconditional abc-substitutes (Stewart-Tijdeman log c << R^{1/3}(log R)^3) are too weak. The missing input is an explicit absolute K0 with d^2 q^3 <= K0 for every #366 case (abc-strength for this family). Sharpened obstruction map; not a proof, no witness.","grade":"obstruction_map","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[{"id":"A060355","name":"Numbers k such that k and k+1 are powerful numbers.","terms":"8,288,675,9800,12167,235224,332928,465124,1825200,11309768,384199200,592192224,4931691075,5425069447,13051463048,2213222","url":"https://oeis.org/A060355"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}