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| 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.
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| Usually, this involves 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, base ten is assumed.
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| 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 is the measure of how many times we must [[digit sum|sum the digits]] it takes us to arrive at its digital root.
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| == 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.
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| == Smallest numbers of a given persistence ==
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| For a radix of 10, there is thought to be no number with a multiplicative persistence > 11: this is known to be true for numbers up to 10<sup>50</sup>. The smallest numbers with persistence 0, 1, ... are:
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| :0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... {{OEIS|A003001}} | |
| By cleverly using the specific properties of numbers in this sequence, the above terms can be calculated in a fraction of a second.
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| The additive persistence of a number, however, can become arbitrarily large (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, ... are:
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| :0, 10, 19, 199, 19999999999999999999999, ... {{OEIS|A006050}} | |
| The next number in the sequence (the smallest number of additive persistence 5) is 2 × 10<sup>2×(10<sup>22</sup> − 1)/9</sup> − 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]].
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| ==References==
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| * {{cite book |last=Guy | first=Richard K. | authorlink=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 }}
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| [[Category:Number theory]]
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