Aperiodic tiles with two distinct odd prime factors or higher prime-square cardinality

Determine whether finite translational tiles whose cardinality is a product of two distinct odd primes, or is \(p^2\) for a prime \(p\geq5\), can be aperiodic; equivalently, establish or refute full periodic tileability for these unresolved cardinality families.

Background

The paper proves full periodicity for tiles of cardinality $2q$, with qq an odd prime, and for tiles of cardinality nine. Combined with earlier results for singleton, prime, and four-point tiles, this leaves specific cardinality families with exactly two prime factors, counted with multiplicity, outside the paper’s conclusions.

The unresolved families are products of two distinct odd primes and prime squares p2p^2 for primes at least five. The open issue is whether tiles in these families can admit translational tilings without any fully periodic tiling complement.

References

Thus aperiodic tiles cannot have cardinality $2q$ or $9$; among cardinalities with exactly two prime factors, counted with multiplicity, the unresolved families are products of two distinct odd primes and squares $p2$ with $p\geq5$.

— Periodic Tilings of Cardinality Twice a Prime or Nine in Arbitrary Dimension  (2609.26576 - Tan et al., 22 Sep 2026) in Section 1, Introduction, subsection “Background and main result”