[seqfan] Re: (prime(n)*prime(n+1)) mod prime(n+2) is odd

[Vladimir Shevelev]

Dear Zak, Giovanni and SeqFans,

At the present time it is impossible to
strictly prove that Zak's sequence is 
finite. The best upper estimate for gap g_n
was obtained in 2001 by R. C. Baker, G. Harman, 
J. Pintz (The difference between 
consecutive primes, II, Proceedings 
of the London Mathematical Society,
83 (3), 532–562) ) : g_n<p^0.525.
Even Riemann conjecture gives only
g_n<O(sqrt(p_n)log(p_n)).
It is also not sufficient for proof. Only 
from Cramer conjecture g_n=O((log p_n)^2)
(1936) the proof follows. But it should
be happen a deep revolution in Mathematics
in order to prove it.

Best regards,
Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Giovanni Resta [giovanni.resta at iit.cnr.it]
Sent: 07 June 2017 12:32
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: (prime(n)*prime(n+1)) mod prime(n+2) is odd

On 06/07/2017 01:54 AM, zak seidov via SeqFan wrote:
> (prime(n)*prime(n+1)) mod prime(n+2) is odd.
>  n = 1, 2, 5, 7, 10, 14, 15, 23, 29, 46, 61.
> Finite? Full?

Probably finite and full.

If you write
prime(n)=p, prime(n+1)= p + g and prime(n+2)=p+g+h,
(i.e., g and h are the gaps between prime(n), prime(n+1), and prime(n+2),

then
the quotient of p(p+g) divided by p+g+h is
p-h and the remainder is g*h+h^2.

In general the value of g*h+h^2 is even.

To be odd, we need that g*h+h^2 > p+g+h=prime(n+2), so that
the real remainder is not g*h+h^2 but
g*h+h^2 - prime(n+2), which is odd.

But this could reasonably happen only when the primes involved
are small, and thus the gaps between them are large in comparison.

When the primes are larger, we know that, very roughly speaking, we can
expect the gaps to be about O((log p)^2), so the expression g*h+h^2
can't be larger than prime(n+2).

I'm not up-to-date about results on prime gaps, so I don't know
if this can be proved. Probably it can be shown it is implied
by one of the many conjectures about prime gaps.

Giovanni


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