[math-fun] Euclid numbers

[Richard Guy]

Little hope of answering Dan's questions in the
foreseeable future.

Could someone check this against  UPINT3  A2
the beginning of which is quoted below.

The primes listed at    p# + 1   comprise
sequence  A005234.  A reference to UPINT A2 would
be in order.

Their ranks comprise sequence  A014545.  I checked
this against Abramowitz & Stegun, except for the
last three entries.  This is the sequence partially
quoted by Dan.  The sequence number he mentions,
A006862, is not the same.    ^^^^^^

Does anyone know of additional entries to any of
these sequences?  (I'm working on UPINT4 :-)   R.
-----------------
\usection{A2}{Primes connected with factorials.}

\hGidx{factorial $n$}
Are there infinitely many primes of the form $n!\pm1$
or of the form $p\#\pm1$, where $p\#$ is the product,
\Gidx{primorial $p$}, of the primes
$2\cdot3\cdot5\cdots p$ up to $p$.

Discoveries since the second edition by Harvey Dubner
and others have brought the lists to:

$n!+1$ is prime for $n=1$, 2, 3, 11, 27, 37, 41, 73,
77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380.

$n!-1$ is prime for $n=3$, 4, 6, 7, 12, 14, 30, 32,
33, 38, 94, 166, 324, 379, 469, 546, 974, 1963,
3507, 3610, 6917, 21480.

$p\#+1$ is prime for $p=2$, 3, 5, 7, 11, 31, 379,
1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523,
23801, 24029, 42209, 145823, 366439, 392113.

$p\#-1$ is prime for $p=3$, 5, 11, 13, 41, 89, 317,
337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569,
13033, 15877.
--------------------
On Tue, 13 Apr 2004, Dan Asimov wrote:

> Let the nth Euclid number E_n be defined as 1 + (p_1 * ... * p_n)
> (aka 1 + the nth "primorial"), where p_n is the nth prime number.
> 
> Neil Sloane's EIS sequence A006862 lists the first few n for which E_n is
> prime:
> 
>             1,2,3,4,5,11,75,171,172,384,457,616,643,....
> 
> Some questions:
> 
> 1. Are there infinitely many prime (resp. composite) E_n ???
> 
> 2. Is there a nice asymptotic expression for the number of E_n < x ???
> 
> 3. Same for prime (resp. composite) E_n ???
> 
> --Dan
> 
> Daniel Asimov
> Visiting Scholar
> Mathematics Department
> University of California
> Berkeley, California
> 
> 
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