Numbers of complex idempotents

Construct, for every positive integer n and every integer m with $2\le m\le2^n$, a perfect n-dimensional complex evolution algebra having exactly m distinct idempotents, including zero.

Background

The real case is proved: every cardinality from two through 2n2^n occurs. Over the complex numbers, the conjecture remains unresolved beyond the verified low-dimensional cases and would in particular produce a perfect algebra with exactly one non-zero idempotent in every dimension.

References

For every $n$ and every $m\in{2,\dots,2n}$ there is a perfect $n$-dimensional complex evolution algebra with exactly $m$ distinct idempotents (including $0$).

— Evolution algebras: a survey of results and a list of open problems  (2610.22248 - Kaygorodov et al., 15 Oct 2026) in Conjecture c:uniqueidemp, Section 2