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

israel at math.ubc.ca israel at math.ubc.ca
Fri May 3 00:33:57 CEST 2019

```Your conjecture is exactly the Goldbach conjecture. That is, if 2*n = p+q
with p > q, then p = n+k and q = n-k where k = p-n.

Cheers,
Robert

On May 2 2019, Peter Luschny 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