Persistence of a number
Encyclopedia
In mathematics
, the persistence of a number is a term used to describe the number of times one must apply a given operation to an integer before reaching a fixed point
, i.e. until further application does not change the number any more.
Usually, this refers to the additive or multiplicative persistence of an integer, which is how often one has to replace the number by the sum or product of its digits until one reaches a single digit. Because the numbers are broken down into their digits, the additive or multiplicative persistence depends on the radix
. In the remaining article, we will assume a radix of 10.
The single-digit final state reached in the process of calculating an integers's additive persistence is its digital root
. Put another way, a number's additive persistence is the measure of how many times we must sum the digits
it takes us to arrive at its digital root.
By cleverly using the specific properties of numbers in this sequence, the above terms can be calculated in a fraction of a second.
The additive persistence of a number, however, can become arbitrarily large (proof: For a given number
, the persistence of the number consisting of 
repetitions of the digit 1 is 1 higher than that of 
). The smallest numbers of additive persistence 0, 1, ... are:
The next number in the sequence (the smallest number of additive persistence 5) is 2 × 102×(1022 − 1)/9 − 1 (that is, 1 followed by 2222222222222222222222 9's). For any fixed base, the sum of the digits of a number is proportional to its logarithm
; therefore, the additive persistence is proportional to the iterated logarithm
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the persistence of a number is a term used to describe the number of times one must apply a given operation to an integer before reaching a fixed point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
, i.e. until further application does not change the number any more.
Usually, this refers to the additive or multiplicative persistence of an integer, which is how often one has to replace the number by the sum or product of its digits until one reaches a single digit. Because the numbers are broken down into their digits, the additive or multiplicative persistence depends on the radix
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
. In the remaining article, we will assume a radix of 10.
The single-digit final state reached in the process of calculating an integers's additive persistence is its digital root
Digital root
The digital root of a number is the value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum...
. Put another way, a number's additive persistence is the measure of how many times we must sum the digits
Digit sum
In mathematics, the digit sum of a given integer is the sum of all its digits,...
it takes us to arrive at its digital root.
Examples
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 number of multiplicative persistence 3.Smallest numbers of a given persistence
For a radix of 10, there is thought to be no number with a multiplicative persistence > 11. The smallest numbers with persistence 0, 1, ... are:- 0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ...
By cleverly using the specific properties of numbers in this sequence, the above terms can be calculated in a fraction of a second.
The additive persistence of a number, however, can become arbitrarily large (proof: For a given number
, the persistence of the number consisting of repetitions of the digit 1 is 1 higher than that of ). The smallest numbers of additive persistence 0, 1, ... are:
- 0, 10, 19, 199, 19999999999999999999999, ...
The next number in the sequence (the smallest number of additive persistence 5) is 2 × 102×(1022 − 1)/9 − 1 (that is, 1 followed by 2222222222222222222222 9's). For any fixed base, the sum of the digits of a number is proportional to its logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
; therefore, the additive persistence is proportional to the iterated logarithm
Iterated logarithm
In computer science, the iterated logarithm of n, written n , is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1...
.