Ternary expansions of powers of 2

Paul Erdos asked how frequently the ternary expansion of 2^n omits the digit 2. He conjectured this happens only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first is over the real numbers, and considers the integer part of lambda 2^n for a real input lambda. The second is over the 3-adic integers, and considers the sequence lambda 2^n for a 3-adic integer input lambda. We show that the number of input values that have infinitely many iterates omitting the digit 2 in their ternary expansion is small in a suitable sense. For each nonzero input we give an asymptotic upper bound on the number of the first k iterates that omit the digit 2, as k goes to infinity. We also study auxiliary problems concerning the Hausdorff dimension of intersections of multiplicative translates of 3-adic Cantor sets.Comment: 28 pages latex; v4 major revision, much more detail to proofs, added material on intersections of Cantor set

Johann Faulhaber and sums of powers

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber's coefficients can moreover be generalized to factorial powers of noninteger exponents, obtaining asymptotic series for $1^{\alpha}+2^{\alpha}+...+n^{\alpha}$ in powers of $n^{-1}(n+1)^{-1}$