Uniqueness of Singer-prime representations

Characterize all possible representations of a given Singer prime r as (q^d−1)/(q−1), equivalently determine every distinct pair (q,d) that yields the same Singer prime.

Background

A Singer prime may admit more than one representation as (qd−1)/(q−1), and the paper later gives the example 31=(53−1)/(5−1)=(25−1)/(2−1). Determining whether such representations are unique, and classifying all exceptions, is relevant to identifying the proper linear subgroups that arise in the analysis of Hall rotary pairs.

References

Moreover, characterizing all possible representations of a given Singer prime, that is, determining all distinct pairs $(q,d)$ that yield the same prime $r$, remains an interesting and unresolved problem that merits further investigation.

— Regular, and (bi-)rotary Hall Cayley maps  (2608.30196 - Di et al., 31 Aug 2026) in Section 2, following Definition of Singer primes