Persistence of a number assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root. The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest
Sum-product number in a reasonable period of time. From Wilson's proof, Raymond Puzio developed the proof that in any positional base system there is only a finite set of sum-product numbers. First he observed that any number "n" of length "l" must satisfy formula_4. Second, since the largest digit in the base represents "b" - 1, the maximum possible value of the sum of digits of "n" is formula_5 and the maximum possible value of the product of digits is formula_6. Multiplying the maximum possible sum by the maximum possible product gives formula_7, which is an upper bound of the value of