- Apr 14, 2005I have been looking at this paper of Harvey's. I think I have

discovered a few things which are very interesting. First let me

list the numbers which he found not to be middle numbers < 2*10^10:

N = 96, 402, 516, 786, 906, 1116, 1146, 1266, 1356, 3246, and 4206.

Harvey stated to me the correction of 4208 should be 4206.

Note that all except the second one, 402, is == 1 (mod 5). Also as

would be expected all are divisible by 6, but they are a complete

residue set with (mod 4) == (mod 24) (I might be stating this wrong,

so let me restate this when divided by 24 the remainder is one of

0,1,2, or 3.) There is only one of these with == 0 (mod 24), namely

96. There is also a interesting thing with (mod 11) and N/6 == (mod

11) most have a factor of 2 except the first three. I did look at

(mod 7). There are no == 1,4 (mod 7). I wonder how this fits with

http://primepuzzles.net/problems/prob_003.htm ?

N, (mod 11), N/6 == (mod 11), (mod 7)

96, 8, 5, 5

402, 6, 1, 3

516, 10, 9, 5

786, 5, 10, 2

906, 4, 8, 3

1116, 5, 10, 3

1146, 2, 4, 5

1266, 1, 2, 6

1356, 3, 6, 5

3246, 1, 2, 5

4206, 4, 8, 6

Add to these statement a look a MR1745569,

http://primes.utm.edu/references/refs.cgi?author=Suzuki

I have a scaned it if anyone wants a copy.

Because of the above, 4*n+/-1 and 6*m+/-1 are the only ways to state

all primes factors with only two Dirichlet equations, and because

4206 < 6!*6, I don't think there are any more possible. All other

numbers > 6!*6 have some way of having numbers > 36 = 6^2 in the P2

= p1*p2 product (prime factors are of 6*n +/- 0, 1, 2, 3, 4, or 5, n

< 6) + prime as to Chen's theorem (all even = P2 + p3).

I do see how this can be used as part of the proof of the t-prime-GC

if anyone is up to it.

Can anything more be said?

John

--- In primenumbers@yahoogroups.com, Harvey Dubner <harvey@d...>

wrote:> I had a paper published in Journal of Recreational Mathematics,

Vol.30(3),

> 1999-2000, entitled

where a

> "Twin Prime Conjectures" which included the conjecture,

>

> Every even number greater than 4208 is the sum of two t-primes

> t-prime is a prime which has a twin.

the

>

> This was verified up to 4.10^11 and included strong evidence that

> conjecture would hold for larger numbers. The paper also includes

conjecture or GC.

> interestng comparative Goldbach data.

>

> I will be happy to send a pdf copy of the paper to anyone who asks.

>

> Harvey Dubner

>

> PS: No, the paper does not include a proof of the above

> PPS: No, I am not working on such a proof.

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