[seqfan] Re: Does every even number >= 4 have a centered Goldbach partition?

[Hugo Pfoertner]

When trying to find a counterexample, I find the following pair of
sequences:
4-1 and 4+1 both prime -> offset d=1
6-1 and 6+1 both prime -> offset d=1
8-1, 8+1 not both prime, offset d=3 works
8-3, 8+3 both prime
none of 10-3, 10+3; 12-1, 12+1; 14-3, 14+3; 16-3,16+3; 18-1,18+1; 20-3,20+3
needs an offset >3
but for n=2*11=22 the least offset making 2*n-d and 2*n+d both prime is d=9.
22-9 and 22+9 are both prime.
The next even number needing a larger offset is 2*23=46. 46-15 and 46+15
are primes.

It seems that except for n=2 all record producing offsets are divisible by
3: d=3*k

2*n +- 3*k is a "centered" Goldbach pair with record spacing. (better title
needed)
n  k
4 1
11 3
23 5
64 7
68 9
73 11
119 13
143 15
172 25
263 29
452 31
557 39
868 45
1238 61
1579 63
2864 75
3533 81
3637 85
4252 111
5171 123
9263 131
11282 151
12388 175
20036 207
59119 225
69332 241
90131 267
113783 301
139283 329
178612 335
185714 427
413788 435
468059 505
579932 539
960707 565
1879582 611
2727031 617
3266951 665
3319868 695
3591593 739
7550768 805
8323226 843
12171914 875
13536176 1113
23165434 1143
28994678 1465

Hugo Pfoertner

On Thu, May 2, 2019 at 10:54 PM Peter Luschny <peter.luschny at gmail.com>
wrote:

> Let N = 2*n = p + q where p and q are primes.
> We call the pair (p, q) a Goldbach partition of N.
>
> The Goldbach conjecture states that every even integer greater
> than or equal to 4 is the sum of two primes. A065577 shows that
> an integer can have many Goldbach partitions.
>
> A centered Goldbach partition of 2*n is the Goldbach partition
> of the form (n - k, n + k) where the k >= 0 is minimal.
>
> For instance if N is twice a prime than k = 0.
> If N = 18 then k = 2 because (9 - 2, 9 + 2) is a Goldbach partition
> of 18 and neither (9 - 0, 9 + 0) nor (9 - 1, 9 + 1) is one.
>
> By the minimality condition every even number has at most one
> centered Goldbach partition. But does every even number have
> a centered Goldbach partition?
>
> I conjecture that the answer is 'Yes'. Our conjecture is much
> stronger than the Goldbach conjecture. Consequently it should
> be more easy to disprove it.
>
> Who is the first to give a counter-example? If you find one
> please enter it in A325142.
>
> Cheers, Peter
>
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