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

[Robert Gerbicz]

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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