A Counterexample to Conjectures by Sloane and Erdős concerning the Persistence of Numbers

If the digits of any multi-digit number are multiplied together, another number results. If this process is iterated, eventually a single digit number will be produced. The number of steps that this process takes, before a single digit number is obtained, is referred to as the persistence of the of the original number [5]. Neil Sloane conjectured that for any base b, there is a number c(b) such that the persistence in base b cannot exceed c(b). According to Richard Guy [2], Erdős Pàl has made a similar conjecture regarding the persistence of numbers in which only non-zero digits are considered. No doubt both Sloane and Erdös were assuming fixed, or single, radix systems when making their conjectures. Nonetheless, this assumption is not explicitly stated, and if a fixed radix system is not assumed, then the conjectures are false. Readers may recall that in factorial base [4] (also referred to as “factorian”) integers are represented as the sum of multiples of factorials [1][3]. The right-most digit represents multiples of 1!, the next digit to the left represents multiples of 2! and so on. For small numbers it is convenient simply to indicate the factorial base thus, 3710 = (1 × 4!) + (2 × 3!) + (0 × 2!) + (1 × 1!) = 1201F . With larger numbers, and particularly when referring to individual digits of the number, it is easier to show the meaning of each digit explicitly within the representation; thus