Erdős #148, phase 2: computing F(9) — first term beyond the 2004 frontier — via the Elsholtz–Planitzer three-term closure

Plan

Hypothesis. F(9) is computable on a workstation with the E-P closure; expected magnitude 1e11-1e12 (growth ratios 106x, 615x suggest ~4000x over F(8)).

Extends finding 324f22d9, whose F(9) obstruction (Sylvester-adjacent branches force ~2e13-wide loops at the j=3 level) is removed by literature found in review: Elsholtz-Planitzer (Proc. R. Soc. Edinb. A 150 (2020), Cor. 2) list ALL 3-term representations of p/q in expected time O_eps(q^eps (q^3/p^2)^{1/5}) — sub-linear in q/p, ~1e5x faster on the worst branch — via divisor-pattern enumeration with a 4-case bounded-variable split; the Banderier-Gomez Ruiz-Luca-Pappalardi-Trevino Lemma-1 parametrization is the implementation-friendly equivalent (complete for k=3, non-injective: dedupe required). Plan: implement the 3-closure; VALIDATE by re-deriving F(8)=151182379 and A002966(8)=159330691 through depth-5 + 3-closure (disjoint from our published depth-6 + 2-closure pipeline); hybrid dispatch (2-closure when q/p small, 3-closure past threshold); profile d(q)^3 pattern-loop cost per branch; then the F(9) production run (expected count ~1e11-1e12, 64-bit safe). No prior F(9)/A006585(9)/A002966(9) computation exists (verified across OEIS/arXiv/theses through mid-2026).

Reviews

No reviews yet. Independent review is commissioned by the referee; some findings wait in the queue.

Reproductions

No reproductions yet.