# [seqfan] Re: 5, 20, 120, 540, 6480, ...

Allan Wechsler acwacw at gmail.com
Wed Mar 30 21:13:14 CEST 2022

```The observation about the smallness of p(n+1) - p(n) is very convincing to
me, and turns me into a believer in this conjecture, but it feels like it
would be very hard to prove, like the twin prime conjecture (on which Tao
which I have heard no news for a very long time).

This makes me suspect that similar conjectures will be true for other
just-slightly-supralinear sequences.

On Wed, Mar 30, 2022 at 1:29 PM Robert Gerbicz <robert.gerbicz at gmail.com>
wrote:

> Hi !
>
> See:
>
> https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes/
> and from that page: https://arxiv.org/pdf/1603.03720.pdf conjecture 1.1
> what they have in conjecture 1.1 is that for consecutive p1,p2 primes you
> will see p2+p1 is divisible by q more often than p2-q1.
>  [because in the latter case p2==p1==a mod q, while in the other case p2
> and p1 are in different residue classes if q>2].
> You'd still need effective constants on that conjecture's bounds, but at
> least we see why this should be true,
> notice also that p(n+1)-p(n) is "small", so you could prove that the
> product is an integer up to a pretty large bound, just factorize the terms
> using prime up to L, if p(n+1)-p(n)<=L is true for all n<=N.
>
>
> Tomasz Ordowski <tomaszordowski at gmail.com> ezt írta (időpont: 2022. márc.
> 30., Sze, 17:07):
>
> >
> > Let a(n) = Product_{k=1..n} (prime(k+1)+prime(k))/(prime(k+1)-prime(k)).
> > Conjecture: a(n) is an integer for every natural n.
> > Is it known or provable?
> >
> > Best regards,
> >
> > Thomas Ordowski
> >
> >