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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