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.

References

[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

Smugglers isomorphism

A colourful colletion of tablets of the drug popularly known as Ecstacy. Photo: justice.gov

A colourful collection of tablets of the drug popularly known as Ecstacy. Photo: justice.gov

Some years ago, Daniel Reidpath and I were experimenting with the idea of using a self-guided learning algorithm to develop rules for discriminating the profiles of drug-smugglers from those of non-smugglers in a database of visitors to Australia. The criterion variable in which we were interested was dichotomous: coded 0 if a visitor was not a drug-smuggler, and coded 1 if they were. Similarly, the predictor variables we examined were also coded dichotomously. Had the visitor had a sojourn in the Golden Triangle on their way to Australia? Were they under 25 years old? Were they male? — and so forth. And we wanted to check our learning algorithm by testing it on a database of simulated cases in which the functional relationship between the predictor variables and the criterion variable was known, and systematically changed by us.

The problem with the simulations was that the number of functions that we might need to look at grows as a double-exponential function of the number of predictor variables. Specifically, if there is but one predictor variable, then there are four possible functions relating the predictor variable to the criterion variable. If there are 2 predictor variables, then there are 16 different functions; with 3 predictor variables the number of possible functional relationships has grown to 256; and, in general, for n predictor variables there are 22^n. However, many of those functions do not need to be tested separately because there is a great deal of redundancy amongst them, at least from the point of view of the learning algorithm. That is, one or more of the functions are essentially isomorphic. Consider, for example, the fact that whether we code the answer to the question, “Is the visitor male?” as 0 for Yes and 1 for No, or vice versa, is essentially irrelevant. Either coding system would work and we knew that if our learning algorithm were capable of detecting a relationship that was coded in one direction then it would be equally capable of detecting it if it had been coded in the opposite direction. The same logic applies to the criterion variable: the criterion could be coded as 1 for drug-smugglers and 0 for non-smugglers, or vice versa.

Two or more predictor variables

When one considers two or more predictor variables, rather than just a single one, as in the case of the sex of the visitor, the redundancy is even greater. It was irrelevant to us whether, for example, we treated the sex of the visitor as the first predictor variable and their (dichotomized) age as the second variable, or the other way around. When all the redundancies are taken into account, the number of functional relationships against which we needed to test our learning algorithm droped considerably. Instead of 256 functional relationships with three predictor variables, only 14 relationships need to be examined. Similarly, in the case of four predictor variables, the number of functions that need examining is reduced from 65536 to 247. It appears that the number of functions to be examined still grows as a double-exponential of the number of predictor variables, but nonetheless for any n, the number is much less than what it would otherwise be.

If you’re interested in more details of the Smugglers isomorphism, you will find them here.

How many non-isomorphic functions are there under the Smugglers isomorphism

It would be nice to have an analytical solution (i.e., an explicit formula) for the number of non-isomorphic functions on n predictor variables.

Contributors: Mark R. Diamond