In
mathematics, a Smarandache–Wellin number is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
that in a given
base is the
concatenation
In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenat ...
of the first ''n''
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s 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.
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
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific con ...
with 5719 digits ending in 11927, discovered by
Eric W. Weisstein
Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...
in 1998. 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.
[ Retrieved 2011-07-28.]
See also
*
Copeland–Erdős constant The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value, using the modern definition of prime, is approximately
:0.235711131719232931374143… .
The constant is irration ...
*
Champernowne constant, another example of a number obtained by concatenating a representation in a given base.
References
External links
*
*
*
List of first 54 Smarandache–Wellin numbers with factorizationsSmarandache–Wellin primes at ''The Prime Glossary''
*Smith, S. "A Set of Conjectures on Smarandache Sequences." Bull. Pure Appl. Sci. 15E, 101–107, 1996.
{{DEFAULTSORT:Smarandache-Wellin number
Base-dependent integer sequences
Prime numbers