Smarandache–Wellin number explained

In mathematics, a Smarandache - Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache - Wellin numbers are named after Florentin Smarandache and Paul R. Wellin.

The first decimal Smarandache–Wellin numbers are:

2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, ... .

Smarandache–Wellin prime

A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 . The fourth is 355 digits long: it is the result of concatenating the first 128 prime numbers, through 719.^[1]

The primes at the end of the concatenation in the Smarandache–Wellin primes are

2, 3, 7, 719, 1033, 2297, 3037, 11927, ... .

The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:

1, 2, 4, 128, 174, 342, 435, 1429, ... .

The 1429th Smarandache–Wellin number is a probable prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein in 1998.^[2] If it is proven prime, it will be the eighth Smarandache–Wellin prime. In March 2009, Weisstein's search showed the index of the next Smarandache–Wellin prime (if one exists) is at least 22077.^[3]

See also

• Copeland–Erdős constant
• Champernowne constant, another example of a number obtained by concatenating a representation in a given base.

External links

• • • • List of first 54 Smarandache–Wellin numbers with factorizations
• Smarandache–Wellin primes at The Prime Glossary
• Smith, S. "A Set of Conjectures on Smarandache Sequences." Bull. Pure Appl. Sci. 15E, 101–107, 1996.

Notes and References

1. Book: Pomerance , Carl B. . Crandall, Richard E. . Carl Pomerance . Prime Numbers: a computational perspective . Springer . 2001 . 78 Ex 1.86 . 0-387-25282-7 .
2. Rivera, Carlos, Primes by Listing
3. Retrieved 2011-07-28.