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Smarandache Prime


A Smarandache prime is a prime Smarandache number, i.e., a prime number of the form 1234...n. 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 77000 to 200000 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 10^6. 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 boundreference
200Fleuren (1999)
38712E. Weisstein (Mar. 21, 2009)
64728E. Weisstein (Oct. 17, 2011)
77000M. Alekseyev (Oct. 3, 2015)
200000S. Batalov (Oct. 22 2015)
344869The Great Smarandache PRPrime search (Dec. 5, 2016)
1000000S. 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).


See also

Consecutive Number Sequences, Integer Sequence Primes, Smarandache Number, Smarandache Sequences

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References

--. "The Great Smarandache PRPrime search." http://smarandache.ddns.net:1200/server_stats.html.Balatov, S. "Smarandache Prime(s)." http://www.mersenneforum.org/showpost.php?p=412039&postcount=1. Oct. 5, 2015a.Balatov, S. http://www.mersenneforum.org/showpost.php?p=413399&postcount=78. Oct 22, 2015b.Balatov, S. https://www.mersenneforum.org/showpost.php?p=489858&postcount=91. Jun 15, 2018.Fleuren, M. "Smarandache Factors and Reverse Factors." Smarandache Notions J. 10, 5-38, 1999.Sloane, N. J. A. Sequences A007908, A071620, and A176942 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Smarandache Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SmarandachePrime.html

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