Sloane’s conjecture on multiplicative digital root

It is believed that there is no integer has a multiplicative persistence greater than itself, a conjecture put forth in 1973 by Neil Sloane, and that Sloane meant to limit this conjecture to fixed radix bases.

In 1998, Diamond and Reidpath published a factorial base counterexample, by proving that “it is possible to find a number in factorial base of arbitrarily large persistence,” specifically a number of the form


Obviously, this number will have a factorial base multiplicative digital root of n and a persistence of also n, suggesting an upper bound for the desired counterexample.


  • 1 M. R. Diamond, D. D. Reidpath, “A Counterexample to Conjectures by Sloane and Erdos Concerning the Persistence of Numbers”, J. Rec. Math. 29 (1998), 89 - 92.
  • 2 N. J. A. Sloane, “The persistence of a number” J. Rec. Math. 6 (1973), 97 - 98.
Title Sloane’s conjecture on multiplicative digital root
Canonical name SloanesConjectureOnMultiplicativeDigitalRoot
Date of creation 2013-03-22 16:00:45
Last modified on 2013-03-22 16:00:45
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 7
Author CompositeFan (12809)
Entry type Conjecture
Classification msc 11A63
Synonym Sloane-Erdős conjecture on multiplicative digital root