[seqfan] Re: Primes modulo primes

[Christopher Gribble]

In A035095, I do not understand the first comment:
"Note that both the terms of this sequence and differences are primes."
I may be showing myself up but what "differences" does the comment refer to?


Chris Gribble

-----Original Message-----
From: seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu]
On Behalf Of franktaw at netscape.net
Sent: 17 June 2010 5:41 AM
To: seqfan at list.seqfan.eu
Subject: [seqfan] Primes modulo primes

Looking at

http://www.research.att.com/~njas/sequences/?q=id%3AA035095|id%3AA032448

first prime == 1 (mod p) and first prime == -1 (mod p), p the nth 
prime, I was wondering if either sequence has any duplicated values.

Both sequences have entries that imply that they do not: in A035095, 
the formula "a(n) is the smallest prime such that greatest prime 
divisor of a(n)-1 is p(n), the n-th prime"; and in A032448, the comment 
"A006530(a(n)+1)=A000040(n)". However, it is not obvious to me that 
these comments are provably correct.

A sufficient condition for them to be correct is that the sequence kp+1 
(resp. kp-1) always has a prime element for some k <= 2p. It appears 
that there is in fact always a prime element for some k <= p. Is this a 
known result? If so, is there a suitable link or reference that can be 
added?

(If the same value P occurred for two primes p and q, it would have to 
be of the form 2kpq+1 (or 2kpq-1). Taking wlog p < q, this would have 
the greatest prime divisor of P-1 not being p, which would contradict 
the above comments. The condition k <= 2p comes directly from this 
expression. (There are problems with initial conditions here for p = 2 
or 3, but these are trivially dealt with.))

Franklin T. Adams-Watters


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