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

[Bob Selcoe]

Giovanni & Seqfans,

Giovanni, I agree with your overall analysis, with one minor point of 
clarification.  You wrote:

> 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.

I think rather g*h+h^2=r - k(prime(n+2)), where k is odd and the largest 
integer such that  r > k(prime(n+2).  So for example we exclude prime(3)=5 
because k=2, even though r=24 > prime(5)=11.  Same with prime(8)=19 (r=60, 
prime(10)=29).

Perhaps the sequence a(n) = k(prime(n+2) - r = {-1, -1, -2, 1, -7, 
7, -1, -2, 15, -11, 1, 31,...}  would be interesting enough for an OEIS 
entry?   Conjectures might include all odd negatives are n = {1, 2, 5, 7, 
10, 14, 15, 23, 29, 46, 61} with k=1, and all other terms are either 
positive with k=0 (infinite) or even negative with k=2 (finite).

Cheers,
Bob Selcoe

--------------------------------------------------
From: "Giovanni Resta" <giovanni.resta at iit.cnr.it>
Sent: Wednesday, June 07, 2017 4:32 AM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
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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