A Smarandache prime is a prime Smarandache number, i.e., a prime number of the form 
. Surprisingly, no Smarandache primes are known as of
Nov. 2015. Upper limits on the non-appearance of primes are summarized in the
table below. The search from 
to 
was completed by Balatov (2015b), and search of larger
terms is now underway (Great Smarandache PRPrime search). As of Jun. 2018, it
is known that there are no Smarandache primes up to index 
. This is consistent with the expected number of primes
among the first million terms: about 0.6 (E. W. Mayer, Oct 09 2015, in
OEIS A007908).
| upper bound | reference |
| 200 | Fleuren (1999) |
 | E. Weisstein (Mar. 21, 2009) |
 | E. Weisstein (Oct. 17, 2011) |
 | M. Alekseyev (Oct. 3, 2015) |
 | S. Batalov (Oct. 22 2015) |
 | The Great Smarandache PRPrime search (Dec. 5, 2016) |
 | S. Batalov (Jun. 15, 2018) |
If all digit substrings are allowed (so that e.g., concatenating just the 1 from 10, just 10111 from 101112, etc. are permitted), prime digit sequences are known. In particular, such primes are Champernowne-constant primes, the first few of which are 1234567891, 12345678910111, ... (OEIS A176942), which have 10, 14, 24, 235, 2804, 4347, 37735, ... decimal digits (OEIS A071620).
