bibkey: batista2026a398581
authors: David Batista
year: 2026
title: OEIS A398581, separation of the lexicographically first and maximum-z Egyptian fraction solutions
doi: null
url: https://oeis.org/A398581
claim: "For strictly ordered positive solutions of 5/k = 1/x + 1/y + 1/z, the entry conjectures that first/maximum separation occurs only for k congruent to one modulo five."
strata_touched:
  - D5/S3/Arith/Congruence/EgyptianFiveFirstMaximum
license: citation-only
triage: anchor

First and maximum third denominators

The entry was read on 2026-09-10. Its conjecture is one-way. It explicitly rejects the converse and records the example k=11 with first solution (3,9,99) and a maximum-z solution (4,5,220). Its residue-four comment gives an exclusion for k=5q+4, q>=1; that comment does not prove the other non-one residue classes.

The repository proof supplies the general integer comparison for all four excluded residues. It uses positive residual gaps, a decreasing bound before 5x=2k, the strict denominator order after that point, and the impossibility of a square congruent to three modulo four in the residual-eight branch.

Verified locator

url: https://oeis.org/A398581

The full sequence text at https://oeis.org/A398581/internal was retrieved and read on 2026-09-10. NAME specifies the ordered reciprocal equation; COMMENTS states “Conjecture: every term is congruent to 1 modulo 5”, the counterexample to the converse, the k=11 comparison, and the residue-four exclusion. There is no DOI (doi: null). These passages locate the problem, not the proof of the full implication supplied by the Lean module.

Search scope

Read all six directly cross-referenced sequence entries: A257843, A075248, A075249, A075250, A075251, A257839, and the official b398581.txt. No complete proof of this target was found there. A257843 and A257839 concern numerator four; the programs in A075249–A075251 stop after their first solution and cannot establish global maximality. GitHub code searches for A398581 and Egyptian in Lean files returned no hits; arXiv all:A398581 returned zero. This is a bounded search statement, not a global novelty claim.