Editing Smith number (section)

== Properties ==
W.L. McDaniel in 1987 [[mathematical proof|proved]] that there are infinitely many Smith numbers.<ref name=CS383/><ref>
{{cite journal | last = McDaniel | first = Wayne | title = The existence of infinitely many k-Smith numbers | journal = [[Fibonacci Quarterly]] | volume = 25 | issue = 1 | pages = 76–80 | date = 1987 | doi = 10.1080/00150517.1987.12429731 | zbl=0608.10012 }}</ref>
The number of Smith numbers in [[base 10]] below 10<sup>''n''</sup> for ''n'' = 1, 2, ... is given by
: 1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, ... {{OEIS|id=A104170}}.

Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called '''Smith brothers'''.<ref name=CS384>Sándor & Crstici (2004) p.384</ref>  It is not known how many Smith brothers there are. The starting elements of the smallest Smith ''n''-tuple (meaning ''n'' consecutive Smith numbers) in base 10 for ''n'' = 1, 2, ... are<ref>{{Cite web |author=Shyam Sunder Gupta |url=http://www.shyamsundergupta.com/smith.htm |title=Fascinating Smith Numbers }}</ref>
: 4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... {{OEIS|A059754}}.

Smith numbers can be constructed from factored [[repunit]]s.<ref>Hoffman (1998), pp. 205–6</ref>{{verify source|date=May 2023}} {{as of|2010}}, the largest known Smith number in base 10  is
:9 × R<sub>1031</sub> × (10<sup>4594</sup> + 3{{e|2297}} + 1)<sup>1476</sup> {{e|3913210}}
where R<sub>1031</sub> is the base 10 [[repunit]] (10<sup>1031</sup> − 1)/9.{{cn|date=May 2023}}{{update inline|date=February 2023}}