Editing Persistence of a number (section)

== 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>