Multiplicative persistence: new null results

Multiply together all the digits of a positive integer, n. Using the result, repeat the digit-multiplication process to obtain a new result. Continue until a single digit result is obtained. The number of steps, p, that it takes for n to be changed to the single digit end-point, is called the multiplicative persistence of n. (Sloane, 1973).

It is widely believed that there is no integer in base-10 that has a multiplicative persistence greater than 11. On several websites there is appears an assertion to the effect that any number with multiplicative persistence greater than 11 must have more than 3000 decimal digits—see, for example, this page at the University of Waterloo. However, I have been completely unable to trace the origin of the assertion. The Mathworld entry on multiplicative persistence refers to the work of Phil Carmody in 2001, and, more modestly, states that the lower bound on a number with persistence greater than 11 is 10233. That provides the background to what follows here.

I used Mathematica to extend the range of known results by testing the multiplicative persistence of all those numbers that can be represented as strings of up to one thousand 2s, one thousand 3s, and one thousand 7s, or as strings of up to one thousand 3s, one thousand 5s, and one thousand 7s. To see why these numbers are of particular interest, see the main paper.

My main result is the null result. None of the integers that I tested, other than those already known, had a multiplicative persistence greater than or equal to 11. Of course, if there is already a genuine basis for the assertion regarding a lower bound of 103000 on a persistence 12, then I have unnecessarily wasted a few hours of computer time!

Almost all of the 1,000,000,000 numbers that I tested (products of powers of 2, 3, and 7, and products of powers of 3, 5, and 7) had a persistence of 2. Put another way, most numbers represented by strings of up to one thousand 2s, one thousand 3s, and one thousand 7s, or by strings of up to one thousand 3s, one thousand 5s, and one thousand 7s, have persistence 3.

An ASCII file of the complete results (omitting those power products with persistence 2) can be downloaded from here. The file is surprisingly small. The rows represent numbers of the form 2j × 3k × 7l and of the form 3j × 5k × 7l. The exponent of 2 is in columns 1–4, column 5 is blank, the exponent of 3 is in columns 6–9, column 10 is blank, the exponent of 5 is in columns 11–14, column 15 is blank, the exponent of 7 is in columns 16–19, column 20 is blank, and the multiplicative persistence of the number formed as the product of the powers of 2, 3, 5 and 7 is shown in columns 21–24. You can also download a Mathematica notebook that produces the results.


[1] Sloane, N J. A. (1973). The persistence of a number. Journal of Recreational Mathematics 6, 97–98.
[2] Diamond, M. R., & Reidpath, D. D. (1998). A Counterexample to Conjectures by Sloane and Erdos Concerning the Persistence of Numbers, Journal of Recreational Mathematics 29, 89–92.

Contributors: Mark R. Diamond