On the impossibility of certain (n2 + n + kn+1) configurations

Philbrook, Jackson; Peet, Benjamin

doi:10.2140/involve.2026.19.297

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

On the impossibility of certain $({n^2+n+k}_{n+1})$ configurations

Jackson Philbrook and Benjamin Peet

Vol. 19 (2026), No. 2, 297–310

DOI: 10.2140/involve.2026.19.297

Abstract

We investigate the impossibility of certain $({n^{2} + n + k}_{n + 1})$ configurations. Firstly, for $k = 2$ , the result of Gropp [9] that $\frac{n^{2} + n}{2}$ is even and $n + 1$ is a perfect square or $\frac{n^{2} + n}{2}$ is odd and $n - 1$ is a perfect square is reproved using the incidence matrix $N$ and analyzing the form of $N^{T} N$ . Then, for all $k$ , configurations where parallelism is a transitive property are considered. It is then analogously established that if $n \equiv 0$ or $n \equiv k - 1 mod k$ for $k$ even, then $\frac{n^{2} + n}{k}$ is even and $n + 1$ is a perfect square or $\frac{n^{2} + n}{k}$ is odd and $n - (k - 1)$ is a perfect square. Finally, the case $k = 3$ is investigated in full generality.

Keywords

configuration, incidence matrix

Mathematical Subject Classification

Primary: 05B20, 05B30

Milestones

Received: 24 April 2024

Revised: 27 October 2024

Accepted: 19 November 2024

Published: 8 March 2026

Communicated by Chi-Kwong Li

Authors

Jackson Philbrook
Department of Mathematics St. Martin’s University Lacey, WA United States

Benjamin Peet
Department of Mathematics St. Martin’s University Lacey, WA United States

Vol. 19, No. 2, 2026