On Sloane’s Persistence Problem

Abstract

We investigate the so-called persistence problem of Sloane, exploiting connections with the dynamics of circle maps and the ergodic theory of actions. We also formulate a conjecture concerning the asymptotic distribution of digits in long products of finitely many primes whose truth would, in particular, solve the persistence problem. The heuristics that we propose to complement our numerical studies can be considered in terms of a simple model in statistical mechanics.

Keywords::

• persistence
• circle maps
• ergodic -actions
• Primary 37E10
• Secondary 37A45
• 37A15
• 11K16

Notes

¹What we call persistence in this paper is sometimes referred to as multiplicative persistence elsewhere, to distinguish it from the similarly defined concept of additive persistence, introduced in [Hinden 74]. Since we will consider only multiplicative persistence, we will have no use for the modifier.

²It is not necessary to use the highly nontrivial result about the full Catalan’s conjecture (concerning the Diophantine equation a^x − b^y = 1; see [Mihăilescu 04]). The special case needed here (with bases 2 and 3, which are prime) can be proved by elementary means; see [LeVeque 77, p. 85].

³There are some analogies between Sloane’s question, at least for base 3, and the Collatz conjecture as discussed in [Tao 11]. In both cases, one represents the problem by a question about a map on a compact abelian group (here the circle, while one uses the dyadics for the Collatz conjecture). Also, in both cases, one can formulate a main aspect of the problem in terms of powers of 2 in base 3, here by expressing that 2ⁿ has at least one zero in its base-3 expansion for all n > 15, and as explained in [Tao 11] for the Collatz case.

⁴Warning: The word “locally” is used here with a different meaning from that used in the study of, e.g., cellular automata; in that other context, the carryover is indeed a very nonlocal effect.

⁵This conjecture is indeed more general than its elementary counterpart given earlier: the latter corresponds to the cases in which the initial sequence of carryovers in the MA is identically zero.

⁶It is an old unresolved question whether the sequence of repunits 1, 11, 111, 1111, 11111, …  written in the decimal system, say, contains infinitely many primes.

⁷Here, ‖ · ‖ is the max-norm: if , then .

⁸Note that the total number of k-blocks is 3^k.