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

[Allan Wechsler]

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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> >
>
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