## Re: (p[n+2]-p[n+1])(p[n+1]-p[n])/4

Expand Messages
• I list the value (p[n+2]-p[n+1])(p[n+1]-p[n])/4 and the first prime p (n+1) past 13*10^10 that verfies that that value shouldn t be on the list, (note: I am
Message 1 of 22 , Oct 1, 2003
I list the value (p[n+2]-p[n+1])(p[n+1]-p[n])/4 and the first prime p
(n+1) past 13*10^10 that verfies that that value shouldn't be on the
list, (note: I am listing the middle prime)

83, 130001990611
89, 130006576939
113, 130007632063
158, 130001684287
194, 130004079547

--- In primenumbers@yahoogroups.com, "Zak Seidov" <seidovzf@y...>
wrote:
> Just sent to OEIS:
>
> 4,7,10,13,16,19,22,25,28,31,34,37,40,
> 43,46,49,52,55,58,61,64,67,70,73,76,
> 79,82,83,85,88,89,91,94,97,100,101,
> 103,106,107,109,112,113,115,118,121,
> 124,127,130,131,133,136,137,139,142,
> 145,148,149,151,154,157,158,160,163,
> 166,167,169,172,173,175,178,179,181,
> 184,187,190,191,193,194,196,197,199
>
> Numbers which can not be of form
> (p[n+2]-p[n+1])(p[n+1]-p[n])/4.
>
> Some cases are proved easily,
> some are only guesses checked
> for n<=2,000,000.
>
> SHANA TOVA!!
>
> Zak
• ... I hope you haven t. I don t really think the Online Encyclopedia of Integer Sequences was created for guesses -- your sequence doesn t belong there. ...
Message 2 of 22 , Oct 1, 2003
--- In primenumbers@yahoogroups.com, "Zak Seidov" <seidovzf@y...> wrote:
> Just sent to OEIS:

I hope you haven't. I don't really think the Online Encyclopedia of Integer Sequences was created for "guesses" -- your sequence doesn't belong there.

> 4,7,10,13,16,19,22,25,28,31,34,37,40,
> 43,46,49,52,55,58,61,64,67,70,73,76,
> 79,82,83,85,88,89,91,94,97,100,101,
> 103,106,107,109,112,113,115,118,121,
> 124,127,130,131,133,136,137,139,142,
> 145,148,149,151,154,157,158,160,163,
> 166,167,169,172,173,175,178,179,181,
> 184,187,190,191,193,194,196,197,199
>
> Numbers which can not be of form
> (p[n+2]-p[n+1])(p[n+1]-p[n])/4.

All numbers of the form 3n+1 are part of the set, and the rest aren't. I'm sure there are people on the list who can give you some more hints if you need them, so I'll just say "k-tuple conjecture."

R. A. Twain
• ... All of that is lost (unless, Zak, you want to cut and paste and post ... Also I did not receive this! ... message, ... a ... post ... is ... the
Message 3 of 22 , Oct 1, 2003
>
All of that is lost (unless, Zak, you want to cut and paste and post
> for me).

Also I did not receive this!

> Shoot, last Friday I sent this fairly long response to this
message,
> which isn't posted here, because this is the only yahoo group I am
a
> part of for which "reply" defaults to the poster and not the whole
> group.
>
> Sigh.
>
> All of that is lost (unless, Zak, you want to cut and paste and
post
> for me).
>
> Anyway, in that intended post I reasoned that the +1 mod 3 residue
is
> impossible, and that the other residues should happen, if searched
> deep enough. As a proof of concept I list the following p values
> that have [p(n+2)-p(n+1)]*[p(n+1)-p(n)]/4=194:
>
> p(n) from {1374538987513, 1374682325293, 1374446799889,
> 1374488450563, 1374402345229, 1374600127909, 1374415063363,
> 1374735847123, 1374546494143, 1374429446293, 1374472611199,
> 1374810758323, 1374520532653, 1374801030553, 1374811265683,
> 1374679474459, 1374724625689, 1374637137469, 1374748849219,
> 1374445952569, 1374505434703, 1374691452103, 1374702725089,
> 1374770337163, 1374598127599, 1374757434889, 1374584932489,
> 1374696888559, 1374406298959, 1374724738573, 1374429655633,
> 1374721538119, 1374775427743, 1374475007509, 1374593340163,
> 1374523688173, 1374670818133, 1374665432893, 1374734529673,
> 1374786320833, 1374515455399}
>
> I believe that many, if not all, of the other -1 mod 3 numbers on
the
> list would also 'fall' to a search of sufficient depth.
>
>
> --- In primenumbers@yahoogroups.com, "Zak Seidov" <seidovzf@y...>
> wrote:
> > Just sent to OEIS:
> >
> > 4,7,10,13,16,19,22,25,28,31,34,37,40,
> > 43,46,49,52,55,58,61,64,67,70,73,76,
> > 79,82,83,85,88,89,91,94,97,100,101,
> > 103,106,107,109,112,113,115,118,121,
> > 124,127,130,131,133,136,137,139,142,
> > 145,148,149,151,154,157,158,160,163,
> > 166,167,169,172,173,175,178,179,181,
> > 184,187,190,191,193,194,196,197,199
> >
> > Numbers which can not be of form
> > (p[n+2]-p[n+1])(p[n+1]-p[n])/4.
> >
> > Some cases are proved easily,
> > some are only guesses checked
> > for n<=2,000,000.
> >
> > SHANA TOVA!!
> >
> > Zak
• Shalom R. A. Twain, 1) OEIS assumes scientific level or smth like, according which you may reject most of seqs in it... 2) guesses in any sci sense are
Message 4 of 22 , Oct 1, 2003
Shalom R. A. Twain,

1) OEIS assumes "scientific level" or smth like,
according which you may reject most of seqs in it...
2) "guesses" in any sci sense are more important
than "established facts"
3) my this particular seq was labeled as "bad sequence"
(see copy of Neil's message -
sorry if someone reads it twise or trice)
and I guess that it will be removed from OEIS
4)in my (weakest) exuse -
i am only amateur and fan
may i only add: "extra" knowledge
6)My deepest respects to all NT gurus
7)Zak

%%%%%%%%%%%%%%%%%%%%end of copy of Neil's message%%%%%%%%%%%%
From: N. J. A. Sloane [njas@...]
Sent: 1:34 02/10/03
this just came in from a correspondent:
ID Number: A087656
URL: http://www.research.att.com/projects/OEIS?Anum=A087656
Sequence: 4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,
<skip>
Name: Numbers that can not be of form (p[n+2]-p[n+1])(p[n+1]-p
[n])/4, where p[i] is i-th prime.
Comments: Some cases are proved easily, some are only guesses checked
only up n<=2000000.
A087657 A087658 A087659
Keywords: nonn,new
Offset: 4
Author(s): Zak Seidov (seidovzf(AT)yahoo.com), Sep 26 2003
>>>

The sequence is bogus because:

1) All numbers of the form 3n+1 (>1) are in it
2) Subject to heuristically safe conjecture, no number not of the form
3n+1 can be in it.

In more detail -

(1) is true as 12n+4 factors into either(3a+1)(3b+1) or (3a+2)(3b+2),
and
neither of those can be the fingerprint for an admissible 3-tuple
apart
from {3,5,7} -> (2)*(2)/4 = 1.

(2) is probably true because any {0,2,6n+2} is an admissible 3-tuple,
and
using the same heuristics as are behind the k-tuple conjecture, but
based
on the principles behind Dirichlet's Theorem. An arithmetic
progression
a+i.b can be created with known composites at a+i.p+1, and a+i.p+
{3..6n+1}
using the chinese remainder theorem. This AP should have a density of
prime k-tuples in proportion (by a fixed constant, realted to the
multiplier b) with the number of tuplets that arbitrary integers would
yield - which by Hardy & Littlewood's (2nd) conjecture is infinite.

e.g. the first number not of the form 3n+1 in the list, 83 fails
because

(00:34) gp > p=257987875972449177033341526073139
257987875972449177033341526073139
(00:35) gp > isprime(p)
1
(00:35) gp > q=nextprime(p+1)
257987875972449177033341526073141
(00:35) gp > r=nextprime(q+1)
257987875972449177033341526073307
(00:35) gp > (r-q)*(q-p)/4
83

(not the smallest, I made a typo in my script, but a number jumped out
within seconds anyway!)
%%%%%%%%%%%%%%%%%%%%end of copy of Neil's message%%%%%%%%%%%%

--- In primenumbers@yahoogroups.com, "ratwain" <ratwain@y...> wrote:
> --- In primenumbers@yahoogroups.com, "Zak Seidov" <seidovzf@y...>
wrote:
> > Just sent to OEIS:
>
> I hope you haven't. I don't really think the Online Encyclopedia
of Integer Sequences was created for "guesses" -- your sequence
doesn't belong there.
>
> > 4,7,10,13,16,19,22,25,28,31,34,37,40,
<skip>
> >
> > Numbers which can not be of form
> > (p[n+2]-p[n+1])(p[n+1]-p[n])/4.
>
> All numbers of the form 3n+1 are part of the set, and the rest
aren't. I'm sure there are people on the list who can give you some
more hints if you need them, so I'll just say "k-tuple conjecture."
>
> R. A. Twain
Your message has been successfully submitted and would be delivered to recipients shortly.