[seqfan] Re: Question on primes

[franktaw at netscape.net]

I think it's hard; I'm also pretty sure it's true.

Up to 10^8, there are no prime gaps big enough that (p+1)*(q-1) <= 
((p+q)/2 - 1)^2.

To violate this, one would need to have q >= p + 4 + sqrt(8*(p+1)).  
Prime gaps on the order of sqrt(p) are generally believed not to exist.

Franklin T. Adams-Watters


-----Original Message-----
 From: Creighton Kenneth Dement 
<creighton.k.dement at mail.uni-oldenburg.de>

Dear Seqfans,

I have a quick question that is probably either trivial or really hard.

Observe that (2 + 1)(13 - 1) = 36 is a square. 2 and 13 are primes, but
they are not consecutive (i.e. there are other primes inbetween).

If p and q are consecutive primes, do we have

(p + 1)*(q - 1) is a square if and only if q = p + 2, i.e. p and q are
twin primes?

Sincerely,
Creighton