The prime power conjecture for perfect difference sets

Let $D\subset \mathbb{Z}/m\mathbb{Z}$ be a perfect difference set if every nonzero $a\in\mathbb{Z}/m\mathbb{Z}$ can be written uniquely as the difference of two elements of $D$ . A perfect difference set has order $n$ when $m=n^2+n+1$ and $|D|=n+1$ . The prime power conjecture. An integer $n\geq 2$ is the order of a perfect difference set if and only if $n$ is a prime power.

Singer constructed perfect difference sets of every prime power order, and the conjecture asserts that no other orders occur. It has been verified computationally for all $n$ up to $2$ billion, but remains open in general.

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The prime power conjecture for perfect difference sets
A perfect difference set of size $q$ is a subset with the property that every nonzero group element has a unique representation as a difference of two elements of the set. Prime Power Conjecture. A perfect difference set of size $q$ exists if and only if $q-1$ is a prime power.

Singer constructed perfect difference sets when $q-1$ is a prime power. The converse, asserting non-existence for all other sizes, is a longstanding open problem in number theory and is used in the paper to connect perfect difference sets with rainbow edge-colorings.
source: Maria Axenovich and Felix Christian Clemen, “Rainbow Subgraphs in Edge-colored Complete Graphs – Answering two Questions by Erdős and Tuza”, arXiv:2209.13867 (2022).
The prime power conjecture for perfect difference sets
A perfect difference set of order $q$ is a set of $q+1$ integers whose pairwise differences, taken modulo $q^2+q+1$ , give all nonzero residue classes. The prime power conjecture. A perfect difference set of order $q$ exists only if $q$ is a prime power.

The conjecture is known in many special cases, including by the Bruck–Ryser and Wilbrink results, and has been verified for $q<2\cdot 10^9$ . It is equivalent to the assertion that every finite cyclic projective plane has prime-power order; the general case remains open.
source: Johan Andersson, “On the solutions to a power sum problem”, arXiv:math/0609621 (2006).

Sources & referencesView supporting material

Primary source

Sarah Peluse, “An asymptotic version of the prime power conjecture for perfect difference sets”, arXiv:2003.04929 (2023).

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The prime power conjecture for perfect difference sets

Equivalent formulations 2