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

Peter Luschny peter.luschny at gmail.com
Thu May 2 22:53:40 CEST 2019

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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