An -persistent number is a Positive Integer which contains the digits 0, 1, ..., 9, and for which , ..., also share this property. No -persistent numbers exist. However, the number is 2-persistent, since but , and the number is 18-persistent. There exists at least one -persistent number for each Positive Integer .
See also Additive Persistence, Multiplicative Persistence
Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 15-18, 1991.