A Smarandache prime is a prime Smarandache number, obtained by concatenating the decimal representations of the first 
positive
integers for some 
.
As of Oct. 2026, no Smarandache primes are known. Reported search limits are
summarized in the table below. Batalov (2015b) extended the search from index 
to 
by Oct. 22, 2015. The subsequent distributed Great
Smarandache PRPrime search reached index 
by Dec. 5, 2016. Batalov (2018) reported on Jun. 15,
2018 that no Smarandache primes occur through index 
. WraithX (2025) subsequently reported no probable
primes through index 
on Mar. 23, 2025.
Mayer (2015) gave a corrected heuristic estimate of about 1.13 primes among the first million terms. His model assigns an eligible term of size 
the probability 
of being prime, conditioning
on its being relatively prime to 30. If the count
is additionally modeled by a Poisson distribution
with this expected count, the probability of finding
no primes is approximately 
.
| search limit | 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) |
 | WraithX (Mar. 23, 2025) |
If initial digit prefixes ending inside the final integer are allowed (for example, ending after the 1 in 10), prime prefixes 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).
