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 any sum-product number of length "l". This suggests that formula_8, or dividing both sides, formula_9. Puzio then deduced that, because of the growth of exponential function, this inequality can only be true for values of "l" less than some limit, and thus that there can only be finitely many sum-product numbers "n". In Roman numerals, the only sum-product numbers are 1, 2, 3, and possibly 4 (if written IIII). Sum-product number A sum-product number is an integer that in a given base is equal to the sum of its digits times the product of its digits. Or, to put it