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Metadata 
bibkey: wu2026a399755
authors: Chai Wah Wu
year: 2026
title: "OEIS A399755: Numbers k such that the generalized Pell equation x^2-(k^2+1)*(y^2+1) = 0 has more than one fundamental solution in (x, y)."
doi: null
url: https://oeis.org/A399755
claim: "Conjecture: A399491(k) < 4*k^3+3*k if and only if k is a term of this sequence."
strata_touched:
  - D5/S3/ArithUnits/PellClassSuccessor
license: citation-only
triage: anchor

Wu’s A399755 conjecture

The saved entry attributes A399755 to Chai Wah Wu, September 10, 2026. Its exact name is:

Numbers k such that the generalized Pell equation x^2-(k^2+1)*(y^2+1) = 0 has more than one fundamental solution in (x, y).

Its named conjecture is:

Conjecture: A399491(k) < 4k^3+3k if and only if k is a term of this sequence.

The companion A399491 name is:

The smallest integer m > n such that (n^2+1)*(m^2+1) is a perfect square.

A399491 is attributed to Zhining Yang, August 30, 2026. Its September 10 comment by Chai Wah Wu states the one-class implication for equality with the cubic bound and conjectures the converse. The companion defines the least partner, whereas A399755 tests membership of k in the set of parameters with multiple generalized solution classes. It is not a claim about the kth enumerated value of A399755.

Mathematical scope

For natural k>0, write D=(k:Int)^2+1. The source equation is x^2-Dy^2=D on arbitrary integral pairs. The class relation is multiplication by an integral norm-one unit: for integer u,v with u^2-Dv^2=1, the pair (r,s) is sent to (ru+Dsv, rv+s*u). This is the relation documented by Robertson in D5/L/robertson2004generalizedpell, pages 12-14. It includes simultaneous negation via (-1,0). Neither positive-only solutions, primitive solutions, squarefree D, nor fractional maximal-order units are part of this model.

More than one fundamental solution means more than one generalized solution class, expressed by two inequivalent integral norm-D solutions. This differs from the fundamental positive solution of the norm-one equation. The entry’s diop_DN program and the versioned SymPy docstring’s one-tuple-per-class convention support this terminology; software execution is not trusted mathematical evidence.

The exact target combines existence of a natural m with k<m, IsSquare((k^2+1)*(m^2+1)), and the minimum condition that every natural n>k satisfying the same square predicate has m<=n. For that least m it asserts the strict cubic equivalence to two inequivalent integral solutions. The quantifier is all k>0. The source conjecture remains a conjecture in this citation; its statements and programs are not premises proving that target. The full model and target are in Problems/oeis-a399755-pell-class-successor.md.

The full A399755 JSON is revision 12, source timestamp 2026-09-14T23:49:50-04:00, retrieved with HTTP 200 at 2026-09-15T04:02:11.774431Z, SHA-256 19d73093da6a2e330eb00325720de647203d56f26ec039678dd82a01d72e3488. The companion A399491 JSON is revision 25, source timestamp 2026-09-12T06:07:18-04:00, SHA-256 84ad2705b0e5283ccdd2ecadfbeabcba5f089081fc0cccf95ad82dad8172050f. The A399755 snapshot contains no bibliography entries; its cross-reference is A399491. The companion contains no bibliography entries and links a b-file. The full entries, including comments, examples, programs, and cross-references, delimit this source record.

The bounded search recorded in issue #8016 found no exact resolution in the checked repository, pinned Mathlib, owner-query, or indexed Lean-code scopes. The inspected Mathlib revision is db584cd6d46c92f209a44c0f1c829460d327499d. Exact-text misses do not exclude semantically equivalent results. Related A173962 and A271636 entries were read in that search; A002315 comments, formulas, and bibliography titles were inspected, but linked papers and Robertson’s cited books were not. Accessible Brave queries found no relevant resolution while also missing the fresh entry, so their negative evidence is weak. Later formula/author queries and Google, DuckDuckGo, and arXiv scopes were unavailable through challenges or rate limits. These are the preregistered boundaries, not an exhaustive literature search or coverage of subsequent live changes. No global priority or universal absence of a previous resolution is claimed.

Rights

Citation-only. Attribution and the short defining and conjecture statements are retained for identification and mathematical discussion. No full OEIS dataset or source program is redistributed here. The source-license page was not independently retrieved; this note does not assert a redistribution license. The SymPy link identifies a versioned semantic reference and grants no evidentiary status to execution.

Verified locator

  • Authoritative entry: https://oeis.org/A399755
  • Saved-entry endpoint: https://oeis.org/search?q=id%3AA399755&fmt=json
  • Companion definition: https://oeis.org/A399491
  • Immutable companion mirror: https://github.com/oeis/oeisdata/blob/ba8580c292f2c51dd0dc3ae2f3ba052c7f17c7d3/seq/A399/A399491.seq
  • SymPy 1.14.0 semantic reference: https://raw.githubusercontent.com/sympy/sympy/sympy-1.14.0/sympy/solvers/diophantine/diophantine.py
  • Exact preregistration and search boundary: https://github.com/the-omega-institute/trureturing/issues/8016

The earlier official mirror lacked A399755 and returned 404. The live snapshot identified above fixes the quoted A399755 source; later edits to that entry are outside this record.