# [seqfan] Re: prime conjecture

Dmitry Kamenetsky dmitry.kamenetsky at rsise.anu.edu.au
Wed Dec 29 09:34:43 CET 2010

```I submitted the following as A178916 and added you as author. Feel free to
modify it as you wish:

%I
%S 1,1,1,1,1,1,5,5,1,1,7,1,7,7,1,7,7,1,7,7,7,11,11,1,1,19,1,1,23,11,17,
%T 1,11,1,1,11,17,29,1,1,13,1,13,1,29,11,29,1,13,1,11,11,1,1,17,1,13,17,
%U 29,1,47,13,1,13,19,17,29,1,17,59,1,1,29,1,41,29,1,1
%N Triangular array a(n,k) read by rows: nextprime(k*n!)-k*n!. For 11 and
1<=k<=n there is a prime in the interval [k*n!+1,k*n!+3*n*log(n)^2]. [From
Robert Gerbicz (robert.gerbicz(AT)gmail.com), Dec 28 2010]
%e Triangle begins:
%e 1
%e 1,1
%e 1,1,1
%e 5,5,1,1
%e 7,1,7,7,1
%e 7,7,1,7,7,7
%e 11,11,1,1,19,1,1
%e 23,11,17,1,11,1,1,11
%e 17,29,1,1,13,1,13,1,29
%e 11,29,1,13,1,11,11,1,1,17
%e 1,13,17,29,1,47,13,1,13,19,17
%e 29,1,17,59,1,1,29,1,41,29,1,1
%K nonn,new
%O 1,7
%A Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Dec 29 2010 Robert
Gerbicz (robert.gerbicz(AT)gmail.com), Dec 29 2010

----------------original message-----------------
From: "Robert Gerbicz" robert.gerbicz at gmail.com
To: "Sequence Fanatics Discussion list" seqfan at list.seqfan.eu
Date: Wed, 29 Dec 2010 02:33:09 +0100
-------------------------------------------------

> 2010/12/29 Dmitry Kamenetsky dmitry.kamenetsky at rsise.anu.edu.au
>
>> Hi Robert,
>>
>> Huge thanks for your input! Your bound is much tighter and I really like
>> how
>> you've shown that there is only a finite number of failures. Since
>> 3*n*log(n)^2 is quite small compared to k*n! I wonder if we can use it to
>> find large primes? Either way we should consider how to turn this into a
>> sequence.
>>
>> Cheers,
>> Dmitry
>>
>> ----------------original message-----------------
>> From: "Robert Gerbicz" robert.gerbicz at gmail.com
>> To: "Sequence Fanatics Discussion list" seqfan at list.seqfan.eu
>> Date: Tue, 28 Dec 2010 12:26:38 +0100
>> -------------------------------------------------
>>
>>
>> > 2010/12/28 Dmitry Kamenetsky dmitry.kamenetsky at rsise.anu.edu.au
>> >
>> >> Hello all,
>> >>
>> >> First of all Happy New Year to everyone! Now for some serious stuff. I
>> have
>> >> the following conjecture:
>> >>
>> >> For every n>=1 and k (1> [k*n!+1, k*n!+n*n].
>> >>
>> >>
>> >> I've checked that this conjecture works for all n> it
>> >> for all n. Can anyone help me? Also it would be nice to get a tighter
>> bound
>> >> on the upper range.
>> >>
>> >>
>> >> Sincerely,
>> >> Dmitry Kamenetsky
>> >>
>> >>
>> >>
>> >>
>> >> _______________________________________________
>> >>
>> >> Seqfan Mailing list - http://list.seqfan.eu/
>> >>
>> >
>> > Yes, that is very likely to be true.
>> > My tighter bound: for every n>1 and 1 [k*n!+1,k*n!+3*n*log(n)^2]
>> interval.
>> (this is true for n PARI-GP),
>> >
>> > n!<=n^n so log(n!) From prime number theorem the probability that a
>> random
>> n is prime is about
>> > 1/log(n). So the probability that the above interval contains no prime
is
>> > about (assuming that these numbers are random):
>> > (1-1/(n*log(n)))^(3*n*log(n)^2)~exp(-3*log(n))=1/n^3, we have n
>> choices
>> > for
>> > k value when n is fixed. That means about 1/n^2 probability that we
fail
>> on
>> > n. But sum(n=2,infinity,1/n^2) converges, from this we can expect only
>> > finite number of n,k pairs that my conjecture is false. Obviously it is
>> only
>> > a heuristic.
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>>
>> One possible sequence could be in a triangle way:
> a(n,k)=nextprime(k*n!+1)-k*n! (for n>=1 and 1
> "how you've shown that there is only a finite number of failures"
> As I said this is only a heuristic, not a proof. Let p(n,k)=the
probability
> that [k*n!+1,k*n!+3*n*log(n)^2] contains no prime, then p(n,k)=O(1/n^3),
> so
> sum(n=2,infinity,k=1,n,p(n,k))=sum(n=2,infinity,O(1/n^2)) then use
> http://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma from it the
> probability that infinitely many of them occur is 0, in other words: by 1
> probability we get finite number of failures.
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

```