Editing Persistence of a number

{{Short description|Property of a number}}
In [[mathematics]], the '''persistence of a number''' is the number of times one must apply a given operation to an [[integer]] before reaching a [[Fixed point (mathematics)|fixed point]] at which the operation no longer alters the number.

Usually, this involves additive or multiplicative persistence of a non-negative 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 remainder of this article, [[base ten]] is 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 [[digit sum|sum its digits]] to arrive at its digital root.

== Examples ==
The additive persistence of 2718 is 2: first we find that 2&nbsp;+&nbsp;7&nbsp;+&nbsp;1&nbsp;+&nbsp;8&nbsp;=&nbsp;18, and then that&nbsp;1&nbsp;+&nbsp;8&nbsp;=&nbsp;9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39&nbsp;→&nbsp;27&nbsp;→&nbsp;14&nbsp;→&nbsp;4. Also, 39 is the smallest number of multiplicative persistence&nbsp;3.

== Smallest numbers of a given multiplicative persistence ==
In [[radix|base]] 10, there is thought to be no number with a multiplicative persistence greater than 11; this is known to be true for numbers up to 2.67×10<sup>30000</sup>.<ref name="OEIS 3001" /><ref>{{cite web|url=http://mathworld.wolfram.com/MultiplicativePersistence.html|title=Multiplicative Persistence|author=Eric W. Weisstein|website=mathworld.wolfram.com}}</ref> The smallest numbers with persistence 0, 1, 2, ... are:
:0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899. {{OEIS|A003001}}

The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be in increasing order (with the exception of the second number, 10), and – except for the first two digits – all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits.
Based on these restrictions, the number of candidates for ''n''-digit numbers with record-breaking persistence is only proportional to the [[square (algebra)|square]] of ''n'', a tiny fraction of all possible ''n''-digit numbers. However, any number that is missing from the [[integer sequence|sequence]] above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 30,000 digits if they do exist.<ref name="OEIS 3001">{{Cite OEIS|A003001}}</ref>

==Properties of additive persistence==
* The additive persistence of a number is smaller than or equal to the number itself, with equality only when the number is zero.
* For base <math>b</math> and [[natural number]]s <math>k</math> and <math>n>9</math> the numbers <math>n</math> and <math>n \cdot b^k</math> have the same additive persistence.
More about the additive persistence of a number can be found [https://www.academia.edu/11654065/On_the_additive_persistence_of_a_number_in_base_p here].

==Smallest numbers of a given additive persistence==
The additive persistence of a number, however, can become arbitrarily large ([[mathematical proof|proof]]: for a given number <math>n</math>, the persistence of the number consisting of <math>n</math> repetitions of the digit 1 is 1 higher than that of <math>n</math>). The smallest numbers of additive persistence 0, 1, 2, ... are:
:0, 10, 19, 199, 19999999999999999999999, ... {{OEIS|A006050}}
The next number in the sequence (the smallest number of additive persistence 5) is 2&nbsp;×&nbsp;10<sup>2×(10<sup>22</sup>&nbsp;−&nbsp;1)/9</sup>&nbsp;−&nbsp;1 (that is, 1 followed by 2222222222222222222222 9's). For any fixed base, the sum of the digits of a number is at most proportional to its [[logarithm]]; therefore, the additive persistence is at most proportional to the [[iterated logarithm]], and the smallest number of a given additive persistence grows [[tetration]]ally.

==Functions with limited persistence==
Some [[function (mathematics)|functions]] only allow persistence up to a certain degree.

For example, the function which takes the minimal digit only allows for persistence 0 or 1, as you either start with or step to a single-digit number.

==References==
{{Reflist}}

== Literature ==
* {{cite book |last=Guy | first=Richard K. | author-link=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | pages=398–399 }}
* {{cite book |last=Meimaris | first=Antonios | title=On the additive persistence of a number in base p| publisher=Preprint | year=2015| url=https://www.academia.edu/11654065}}


{{Classes of natural numbers}}

[[Category:Number theory]]