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

[Allan Wechsler]

I am missing something about this problem. Every Goldbach partition of 2n
is of the form (n+k, n-k). Among all the Goldbach partitions of 2n, one of
them must have minimal k. Does this not show that every number 2n that
admits a Goldbach partition admits a centered Goldbach partition?

On Thu, May 2, 2019 at 4:54 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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