## 19682[PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture

Expand Messages
• Nov 7, 2008
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
• Show all 11 messages in this topic