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

Gaming your way to non-trivial results

In 2005, in a posting on Reality War Games, I suggested the possibility that a well equipped military might create an on-line game that recruited gamers from around the world to participate in the simulation of a planned, real-life, military offensive. Something of much the same kind has just been announced in a paper (doi:10.1038/nsmb.2119) published in the journal, Nature Structural and Molecular Biology.

Online players of the protein folding game Foldit were awarded points for producing progressively more accurate models of the protein known as M-PMV retroviral protease. Players scored points according to the energy state of the protein fold that they produced. The lower the energy state of the folded protein, the higher a player’s score. The Foldit players were able to generate models of sufficient quality that it was then possible to build on those models to determine the actual structure of the protein. So, the idea no longer seems so remote that the next massively parallel multi-player online game that you join will actually be a rehersal for World War III.


Khatib, F., Dimaio, F., Foldit Contenders Group, Foldit Void Crushers Group, Cooper, S., Kazmierczyk, M., Gilski, M., Krzywda, S., Zabranska, H., Pichova, I., Thompson, J., Popović, Z., Jaskolski, M., & Baker, D. (2011). Crystal structure of a monomeric retroviral protease solved by protein folding game players. Nature Structural and Molecular Biology, doi: 10.1038/nsmb.2119

Contributors: Mark R. Diamond

The Dodderimeter: helping to prevent falls in the elderly

Falls in the elderly are a significant cause of mobidity.

Falls in the elderly are a significant cause of mobidity.

Children are forever falling over but rarely come to harm. Falls in adults are rare but are frequently catastrophic. Indeed, they are a major cause of morbidity (and consequent mortality) and represent one of the most significant contributors to hospitalizations of the elderly in developed countries.

Fortunately, researchers at the University of New South Wales have developed a device which might help to identify those at greatest risk of falling. The latest incarnation of the device [1], for which I have coined the name, ‘dodderimeter’, measures acceleration along each of the x, y and z space axes and then uses a combination of frequency-domain and time-domain analyses of the signals to predict the likelihood of a fall. It might even become as ubiquitous as the Holter monitor.

[1] Liu, Y., Redmond, S., Wang, N., Blumenkron, F., Narayanan, M., Lovell, N. (2011). Spectral Analysis of Accelerometry Signals from a Directed-Routine for Falls-Risk Estimation. IEEE Transactions on Biomedical Engineering, 99, 1. doi: 10.1109/TBME.2011.2151193. [PubMed]

VIA Strengths—scoring key

Nelson Mandela, who is widely regarded as showing the virtue of forgiveness. Picture:

Nelson Mandela, who is widely regarded as showing the virtue of forgiveness. Picture:

For a measure that has obtained so much publicity, it is remarkable that the VIA Strengths scale has, so far, and as far as I can tell, no published scoring key. In fact, using a variety of search engines, the only places I could find where you can score the scale are various websites, including, and . In the realm of scientific enquiry, the absence of an open scoring-key is remarkable. Even the Beck Depression Inventory has one, despite copyright being claimed in the test itself.

VIA Strengths scale published scientific work

Searching the web for information on the scale turns up thousands of entries, very few of which are related in any serious way to the scientific investigation of the scale. One of the few is the doctoral dissertation of Dennis P. O’Neil, PhD who did his doctoral research at the Department of Psychology and Neuroscience at Duke University. His dissertation reproduces the entire VIA Strengths scale. Working from that, together with the information published by the International Personality Item Pool and the descriptions in the book by Christopher Petersen and Martin Seligman [1], it is possible to determine the structure, and hence the scoring key.

Structure and scoring key for the VIA Strengths scale

The scale is conceptually divided into blocks of 24 questions. Each block has one question relating to each of the 24 character strengths and virtues. Importantly, it turns out that the order, within each block, in which the questions relate to a strength or virtue, is identical across blocks. The full scoring key is described in detail in a new publication [2] in the December 2010 issue of Psychological Reports written by Angela O’Brien-Malone, Rosalind Woodworth and me. The article is available at the D.O.I. link given in the reference.

Help with scoring

Two versions of a spreadsheet for questionnaire scoring are available for download. A version in Open Document Spreadsheet format can be obtained here; and a version suitable for Microsoft Excel can be found here. I shall update and improve the spreadsheets as time permits.


[1] Petersen, C., & Seligman, M. (2004). Character Strengths and Virtues: A Handbook and Classification. Oxford: Oxford University Press.

[2] Diamond, M., O’Brien-Malone, A., & Woodworth, R. J. (2010). Scoring the VIA Survey of Character. Psychological Reports, 107(4), 833-836. DOI: 10.2466/02.07.09.PR0.107.6.833-836

The immoderately long arm of the Psychology Board of Australia

A very long arm

A very (very) long arm.

On 11 November 2010 the Psychology Board of Australia issued Consultation Paper 6: Proposed registration standard—Limited Registration for Teaching or Research. In the paper the Board outlines a definition of psychological practice that was approved by the Australian Health Workforce Ministerial Council for the purpose of determining whether a registered psychologist has engaged in sufficient recent practice to justify the renewal of his or her registration. In the context for which it was approved, the definition is very useful, but the Board uses that description of psychological practice as justification for demanding the registration of every person whose behaviour falls within the ambit of the definition!

The Board’s demand is almost certainly ultra vires, founded on a misunderstanding of the limited powers granted to the Board by the Health Practitioner Regulation National Law. In a paper that Angela O’Brien-Malone circulated on 3 December 2010 to university psychology departments around Australia, we argued that, were the Board able to effect its claim, it would subvert the explicit object and purpose of the Health Practitioner Regulation National Law, have a seriously adverse effect on the discipline of psychology in Australia, and and have a substantive negative effect on the national supply of health practitioners. Our paper, entitled Beyond Its Power and Against the National Interest, can be downloaded here.

Contributors: Mark R. Diamond