# Smarandache-Wellin prime

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Major Section:
Reference
Type of Math Object:
Definition

## Mathematics Subject Classification

### Smarandache-Wellin prime

What is the interest of this type of prime? Why are they studied in the first place?

Alvaro

### Re: Smarandache-Wellin prime

The Ribenboim and the Crandall/Pomerance books are, like, the holy texts of prime hunters, hee hee. Any kind of prime listed in either of those is worthy studying.

Also 'cause they're so hard to find, in any base! In the first 10^355 there might be just four of them... sniff...

### Re: Smarandache-Wellin prime

> The Ribenboim and the Crandall/Pomerance books are, like,
> the holy texts of prime hunters, hee hee. Any kind of prime
> listed in either of those is worthy studying.

To me, that doesn't really answer the question, only pushes it
off. Presumably, if Ribenboim, Crandall, and Pomerance state that these primes are worth studying in their books, they offer some
reason for this statement. What is that reason?

> Also 'cause they're so hard to find, in any base! In the
> first 10^355 there might be just four of them... sniff...

So what? The series gotten by concatenating radix representations
of primes grows rapidly (faster than exponential) so, by the
prime number theorem, one would expect to find few primes in it.
However, that doesn't tell me why this particular series is
interesting since it is hard to find primes in a random rapidly
increasing sequence.

To clarify my understanding of Alvaro's question, I think the point
here is whether the issue of primes in this sequences is related
to some other topic in math or whether there are theorems, or at
least conjectures, involving them. For instance, I would say that
the reason that Mersenne primes are interesting is not because
Mersenne wrote about them but because a. they are relevant to
perfect numbers and b. there are specific theorems for proving
primality or compositeness which only apply to these numbers.
Likewise, the reason primes in arithmetic progressions are
interesting is because of Dirichlet's theorem and L-functions.
Sure, one can investigate primes in any sequence of integers, but
is there any particular reason for looking at Smarandache's and
Wellin's sequence as opposed to any other sequence?