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Singer classes gallery

For each prime power q, Singer's construction gives q + 1 residues modulo q² + q + 1 such that every nonzero residue appears exactly once as a difference of two of them. Multiplying such a set by any t coprime to q² + q + 1 gives another one. Each diagram below is one of the sets this produces, counted once per family of translates and reflections. Every chord is labeled with its two differences.

Counting these diagrams gives φ(q² + q + 1) / 6e for q = pᵉ, where φ is Euler's totient. If a set and its mirror image count as different, the count doubles to φ(q² + q + 1) / 3e. Singer conjectured that formula in 1938; it was proved by Gerald Berman (1953) and by Heini Halberstam and R. R. Laxton (1964). Both sequences are in the OEIS:

• A335866: Singer sets up to translation, φ(v) / 3e.
• A058241: every planar difference set up to rotation and reflection, found by search rather than construction. It agrees with the Singer count (half of A335866) for every size checked. That this holds for every size is still a conjecture; see Lesson 17.