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Smarandache-Wellin Prime

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A Smarandache-Wellin prime is a Smarandache-Wellin number that is prime. Reported prime or probable prime examples are concatenations of the first n=1, 2, 4, 128, 174, 342, 435, 1429, ... (OEIS A046035; Ibstedt 1998, pp. 78-79; Crandall and Pomerance 2005, p. 78) primes. These correspond to concatenations of all primes up to p_n=2, 3, 7, 719, 1033, 2297, 3037, 11927, ... (OEIS A046284), namely

w_1=2
(1)
w_2=23
(2)
w_4=2357
(3)
w_(128)=2357...719
(4)

(OEIS A069151), which have 1, 2, 4, 355, 499, 1171, 1543, 5719, ... (OEIS A263959) decimal digits. The 1171-, 1543-, and 5719-digit examples were reported using probable prime tests (Rivera).

Smarandache-Wellin primes are the subset of constant primes formed from the Copeland-Erdős constant for which the trailing digits correspond to the full (non-truncated) final concatenated prime.

M. Rodenkirch reported a search completed on Nov. 29, 2025 up to 4×10^6 using the program smwsieve/smwsievecl without finding any new primes. Search limits are summarized in the following table.

limitsearcher
1.5×10^6M. Rodenkirch (2016)
2.5×10^6M. Rodenkirch (Aug. 10, 2025)
4.0×10^6M. Rodenkirch (Nov. 29, 2025)

See also

Constant Primes, Copeland-Erdős Constant, Copeland-Erdős Constant Digits, Integer Sequence Primes, Smarandache-Wellin Number

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References

Crandall, R. and Pomerance, C. Problem 1.86 in Prime Numbers: A Computational Perspective, 2nd ed. New York: Springer-Verlag, p. 78, 2005.Ibstedt, H. "Smarandache Concatenated Sequences." Ch. 5 in Computer Analysis of Number Sequences. Lupton, AZ: American Research Press, pp. 75-79, 1998.Numberphile. "The Most Wanted Prime Number." Featuring N. J. A. Sloane. Dec. 15, 2021. https://www.youtube.com/watch?v=vKlVNFOHJ9I.OEIS Foundation. Sequences related to Most Wanted Primes video. https://oeis.org/wiki/Index_to_OEIS:_Section_Mo#MWP.Rivera, C. "Problems & Puzzles: Puzzle 008-Primes by Listing." https://www.primepuzzles.net/puzzles/puzz_008.htm.Rodenkirch, M. "Smarandache-Wellin Primes." Nov. 11, 2025. https://www.mersenneforum.org/node/15124/page7#post1097782.Sloane, N. J. A. Sequences A046035, A046284, A069151, and A263959 in "The On-Line Encyclopedia of Integer Sequences."

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Smarandache-Wellin Prime

Cite this as:

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

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