## [PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture

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• I had problems understanding the original problem formulation so I will try a more formal description. Given two natural numbers a
Message 1 of 11 , Nov 7, 2008
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I had problems understanding the original problem formulation
so I will try a more formal description.
Given two natural numbers a < b, find b-a+1 distinct natural
numbers N_a to N_b such that I +/- N_I is prime for I = a to b.
In other words, for each integer I from a to b, find a prime
pair of form I +/- N such that different N is used each time.

By my hand calculations, if I starts at a=6 then it can at
most go to b=44. It can do that in four ways, with two possible
combinations at I=8,10,12, and two options at I=43.
(I,N_I): (6,1) (7,4) (8,3 or 5) (9,2) (10,7 or 3) (11,6)
(12,5 or 7) (13,10) (14,9) (15,8) (16,13) (17,14) (18,11)
(19,12) (20,17) (21,16) (22,15) (23,18) (24,19) (25,22)
(26,21) (27,20) (28,25) (29,24) (30,23) (31,28) (32,29)
(33,26) (34,27) (35,32) (36,31) (37,30) (38,35) (39,34)
(40,33) (41,38) (42,37) (43,36 or 40) (44,39).

45+/-N is prime for N = 2, 8, 14, 16, 22, 26, 28, 34, 38,
but they are all taken.

Bill Krys wrote:
> ... these are the only prime gaps I can reliably predict
> where and for hong long they occur.

The maximal prime gaps at
http://hjem.get2net.dk/jka/math/primegaps/maximal.htm
can be used to get an upper limit for how large b can be
for a given value of a, based on Mark's argument.
The actual highest value of b may turn out to be lower than
the limit given in this way.

--
Jens Kruse Andersen
• ... Well Jens I want compensation from Bill for what I feel is about 2 months taken off my life trying to do this by hand in the last three hours. I went as
Message 2 of 11 , Nov 8, 2008
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--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
>
> I had problems understanding the original problem formulation
> so I will try a more formal description.
> Given two natural numbers a < b, find b-a+1 distinct natural
> numbers N_a to N_b such that I +/- N_I is prime for I = a to b.
> In other words, for each integer I from a to b, find a prime
> pair of form I +/- N such that different N is used each time.
>
> By my hand calculations, if I starts at a=6 then it can at
> most go to b=44. It can do that in four ways, with two possible
> combinations at I=8,10,12, and two options at I=43.
> (I,N_I): (6,1) (7,4) (8,3 or 5) (9,2) (10,7 or 3) (11,6)
> (12,5 or 7) (13,10) (14,9) (15,8) (16,13) (17,14) (18,11)
> (19,12) (20,17) (21,16) (22,15) (23,18) (24,19) (25,22)
> (26,21) (27,20) (28,25) (29,24) (30,23) (31,28) (32,29)
> (33,26) (34,27) (35,32) (36,31) (37,30) (38,35) (39,34)
> (40,33) (41,38) (42,37) (43,36 or 40) (44,39).
>
> 45+/-N is prime for N = 2, 8, 14, 16, 22, 26, 28, 34, 38,
> but they are all taken.
>
> Bill Krys wrote:
> > ... these are the only prime gaps I can reliably predict
> > where and for hong long they occur.
>
> The maximal prime gaps at
> http://hjem.get2net.dk/jka/math/primegaps/maximal.htm
> can be used to get an upper limit for how large b can be
> for a given value of a, based on Mark's argument.
> The actual highest value of b may turn out to be lower than
> the limit given in this way.
>
> --
> Jens Kruse Andersen
>

Well Jens I want compensation from Bill for what I feel is about 2
months taken off my life trying to do this by hand in the last three
hours. I went as far as cutting out fifty six little pieces of paper
with the numbers from 0 to 55 written on them. Yes, I started at zero
just to shake things up. (The application to Goldbach's conjecture
would still apply, ie, 3 + 0 = 3; 3 - 0 = 3 ; 6 = 3 + 3)

Is this what life was like before computers?

But lo and behold it worketh!:

(6,1) (7,4) (8,5) (9,2) (10,3) (11,6) (12,7)
(13,10) (14,9) (15,8) (16,13) (17,14) (18,11) (19,12)
(20,17) (21,16) (22,15) (23,0) (24,19) (25,22) (26,21)
(27,20) (28,25) (29,18) (30,23) (31,28) (32,29) (33,26)
(34,27) (35,24) (36,31) (37,34) (38,35) (39,32) (40,33)
(41,48) (42,37) (43,40) (44,39) (45,38) (46,43) (47,36)
(48,41) (49,30) (50,47) (51,46) (52,45) (53,50) (54,49)
(55,52) (56,51) (57,44) (58,55) (59,54) (60,53) (61,42)

Mark
• ... Lo and behold it doesn t. Jen noticed that (41,48) yields a negative prime , so is bad. Now, I have to determine if this whole exercise actually caused
Message 3 of 11 , Nov 9, 2008
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<mark.underwood@...> wrote:
>
> But lo and behold it worketh!:
>
>
> (6,1) (7,4) (8,5) (9,2) (10,3) (11,6) (12,7)
> (13,10) (14,9) (15,8) (16,13) (17,14) (18,11) (19,12)
> (20,17) (21,16) (22,15) (23,0) (24,19) (25,22) (26,21)
> (27,20) (28,25) (29,18) (30,23) (31,28) (32,29) (33,26)
> (34,27) (35,24) (36,31) (37,34) (38,35) (39,32) (40,33)
> (41,48) (42,37) (43,40) (44,39) (45,38) (46,43) (47,36)
> (48,41) (49,30) (50,47) (51,46) (52,45) (53,50) (54,49)
> (55,52) (56,51) (57,44) (58,55) (59,54) (60,53) (61,42)
>
>
> Mark
>

Lo and behold it doesn't. Jen noticed that (41,48) yields a negative
'prime', so is bad. Now, I have to determine if this whole exercise
actually caused my mental decline, or whether it was a pre existing
condition.

Mark
• Mark and Jens,   thanks for trying. I am going to withdraw from the group for a while to tend to work, but I ll come back if I find anything or need more
Message 4 of 11 , Nov 12, 2008
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Mark and Jens,

thanks for trying. I am going to withdraw from the group for a while to tend to work, but I'll come back if I find anything or need more help. I'll try to see if either a continuous sequence of "N"s works starting from a higher integer and if no luck there, then I'll see if your idea of sequential fragments works, hopefully based on some easily predictable prime gaps because I don't like the idea of a prime gap I can't predict understand.

P.S. Mark, sorry for causing your cognitive dissonance, but that's learnin', ain't it?

Bill Krys

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--- On Sun, 11/9/08, Mark Underwood <mark.underwood@...> wrote:

From: Mark Underwood <mark.underwood@...>
Subject: [PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture
Date: Sunday, November 9, 2008, 2:57 PM

--- In primenumbers@ yahoogroups. com, "Mark Underwood"
<mark.underwood@ ...> wrote:
>
> But lo and behold it worketh!:
>
>
> (6,1) (7,4) (8,5) (9,2) (10,3) (11,6) (12,7)
> (13,10) (14,9) (15,8) (16,13) (17,14) (18,11) (19,12)
> (20,17) (21,16) (22,15) (23,0) (24,19) (25,22) (26,21)
> (27,20) (28,25) (29,18) (30,23) (31,28) (32,29) (33,26)
> (34,27) (35,24) (36,31) (37,34) (38,35) (39,32) (40,33)
> (41,48) (42,37) (43,40) (44,39) (45,38) (46,43) (47,36)
> (48,41) (49,30) (50,47) (51,46) (52,45) (53,50) (54,49)
> (55,52) (56,51) (57,44) (58,55) (59,54) (60,53) (61,42)
>
>
> Mark
>

Lo and behold it doesn't. Jen noticed that (41,48) yields a negative
'prime', so is bad. Now, I have to determine if this whole exercise
actually caused my mental decline, or whether it was a pre existing
condition.

Mark

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