Cubic and Higher Modulus Borwein Conjectures

Definitions and mathematical statement

The supplied record names several coefficient-sign conjectures but does not supply their defining expressions, coefficient sign patterns, parameter domains, or quantifiers. Consequently, their precise mathematical statements cannot be reconstructed from this record alone.

The named targets must be distinguished as follows. Wang's Cubic Borwein Conjecture is identified as Conjecture 3 in the report's modulus-$3$ family, alongside the First and Second Borwein Conjectures. The Third Borwein Conjecture is a separate conjecture pertaining to modulus $5$. The excerpt also mentions further conjectures of Borwein type for moduli $4$ and $7$, based on computer experiments.

The requested problem, at the level supported by the supplied evidence, is:

Establish the full coefficient-sign assertions of Wang's Cubic Borwein Conjecture and the Third Borwein Conjecture pertaining to modulus $5$, and establish the additional Borwein-type conjectures for moduli $4$ and $7$ mentioned in the report.

This is an unresolved formulation of the intended targets, rather than a complete quantified mathematical statement: the expressions whose coefficients are considered, the asserted signs, all parameter ranges, and any exceptional or boundary cases are missing.

The report claims a uniform proof of the First and Second Borwein Conjectures and a proof of two thirds of Wang's Cubic Borwein Conjecture. It expresses confidence that its approach could prove three fifths of the modulus-$5$ Third Borwein Conjecture, while stating that further refinements would be needed for full proofs of the latter and of Wang's cubic conjecture. The excerpt does not identify the coefficient classes represented by these fractions. Its prospective modulus-$5$ claim must not be treated as an established partial proof.

Short English statement

The intended task is to prove the remaining cubic Borwein sign assertions and the related conjectures for moduli $4$, $5$, and $7$. The supplied excerpt omits the formulas, sign patterns, and parameter ranges needed for a precise mathematical statement. Its partial-proof claims and dated status assessment are supplied source claims requiring review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005322$, title Cubic and Higher Modulus Borwein Conjectures, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2022$ and no named proposer.
• Supplied report citation: Enumerative Combinatorics (2023), Oberwolfach Reports, pp. 3215--3305, DOI 10.4171/owr/2022/57. Supplied source URL and literature-source link labelled original OWR report: https://doi.org/10.4171/owr/2022/57. The supplied passage refers to Conjectures 1--3 and references [6] and [7], whose statements and bibliography entries are absent from the record.
• Supplied literature-source link labelled Wang--Krattenthaler: https://arxiv.org/abs/2201.12415. No paper title or full bibliographic entry is supplied. The record does not explicitly establish its correspondence with either numbered reference in the excerpt.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, reports proofs of the first and second cases and substantial parts of the cubic case, and describes the full cubic and additional modulus conjectures as not established. This is supplied dated curation, not an independently verified assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims. The registry label corrected_verified is accompanied by an explanation limited to repairs of delimiters, notation, grammar, and typography; it does not establish mathematical correctness or completeness.