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

[Tomasz Ordowski]

Primes p such that p+2 divides (p-1)!
are primes p such that q-p does not divide q+p,
where q is the next prime after p.
These are the primes p for which q-p > 2.
Cf. A049579 - OEIS <https://oeis.org/A049579> (see my draft).

Thomas Ordowski

czw., 31 mar 2022 o 05:26 Allan Wechsler <acwacw at gmail.com> napisał(a):

> 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
> and colleagues have also made progress) and the Goldbach conjecture (about
> 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):
> >
> > > Dear readers!
> > >
> > > 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
> > >
> > >
> > > <#m_-7681193058414376069_DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2>
> > >
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