A first digit theorem for powerful integer powers

Abstract

For any fixed power exponent, it is shown that the first digits of powerful integer powers follow a generalized Benford law (GBL) with size-dependent exponent that converges asymptotically to a GBL with the inverse double power exponent. In particular, asymptotically as the power goes to infinity these sequences obey Benford’s law. Moreover, the existence of a one-parametric size-dependent exponent function that converges to these GBL’s is established, and an optimal value that minimizes its deviation to two minimum estimators of the size-dependent exponent is determined. The latter is undertaken over the finite range of powerful integer powers less than $$ 10^{s \cdot m} , \, m = 8, \ldots ,15 $$10s·m,m=8,…,15, where $$ s = 1,2,3,4,5 $$s=1,2,3,4,5 is a fixed power exponent.