[seqfan] Primes modulo primes

franktaw at netscape.net franktaw at netscape.net
Thu Jun 17 06:40:31 CEST 2010

Looking at


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 

(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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