Algebraic Stories — Non-negative forms

Definitions and mathematical statement

The conjecture in the supplied record is the exact identity \[ B'_{n,m}=\left(\frac n2\right)^{m-1} \qquad\text{for every allowed pair $(n,m)$ with even $n$}. \] Here $n$ is even in the sense $n=2r$ for an integer $r$. The record does not define $B'_{n,m}$, the unprimed quantity $B_{n,m}$, the class of non-negative forms being considered, or the admissible ranges and boundary cases of $n,m$. Consequently the full quantifier domain is not recoverable from the displayed statement. In particular, positivity of $n$, a minimum number of variables, and whether $m=0$ or $m=1$ is allowed must not be silently supplied. The formula is retained as the source's proposed exact value, rather than treated as a definition of $B'_{n,m}$.

The supplied research summary reports that this remains a conjecture and quotes the source's separate qualification: "For $B_{n,m}$ no similar guess is known." Thus the primed and unprimed quantities must not be identified. The summary mentions related $O(n^{m-1})$ bounds, but provides no precise bound, parameter regime, or citation; such order-of-growth context does not establish the displayed exact equality.

Short English statement

The source proposes the exact value $(n/2)^{m-1}$ for the primed quantity $B'_{n,m}$ whenever $n$ is even. Its definition and parameter ranges are absent from the supplied record, so a self-contained mathematical statement requires the original non-negative-forms context.

Sources

• Local UnsolvedMath registry, permanent ID 1500019, exact source code catalogue entry, "Algebraic Stories --- Non-negative forms"; status open, classification OPEN-TRIAGE.
• Fröberg, Lundqvist, Oneto, Shapiro, Algebraic Stories from One and from the Other Pockets (2018), source-order Conjecture 19, https://arxiv.org/abs/1801.01692. The record says public arXiv source TeX was inspected, accessed July 29, 2026, and the source presented the item as open.
• The supplied research summary says the paper gives no proof, quotes the absence of a corresponding guess for $B_{n,m}$, and reports that a search suggested the exact formula remains open. It mentions related asymptotic bounds without supplying an independent citation or their hypotheses. These are supplied assessments rather than independently verified literature conclusions.