10:58 AM
Anybody home?

5 hours later…
3:58 PM
You have +1 from me now, even thought I haven't read it yet (I'm always in the middle of something else :P)

4:56 PM
Hmm, I have almost-completed the pattern for this one problem. I have 3 categories of solutions (terms) in the pattern:

1. trivial-cases: Solved: They are 0.
2. general-cases: Solved: They are the sequence of prime gaps!
3. critical-cases: UNSOLVED. I have no idea what pattern these follow?
green is trivial, red and orange is solved (we have a sequence of prime gaps, with offset determined by table row/column), but...

the black region is a mystery i can't figure out

5:08 PM

5:47 PM
(this table extends down and right to infinity)

5:57 PM
The implications of this are... That:
- There are infinitely large numbers that can't be represented as a sum of \$0,1\$ or \$2\$ distinct prime numbers.
- For sufficiently large \$n\$, all numbers \$\ge n\$ can be trivially represented as sums of \$6,7,8,\dots,k-1,\$ or \$k\$ or \$k+1\$ or \$...\$ distinct primes.
- Representing numbers as sums of \$3,4,5\$ distinct primes isn't trivial.

First case of last claim, \$3\$ primes: Goldbach conjecture is that all numbers are a sum of \$3\$ not necessarily distinct primes (it is usually stated just for even numbers as sums of \$2\$ primes).
(I meant "exactly \$k\$ distinct primes" which is not quite equal to goldbach weak/strong conjectures, but is similar)
Even if closed form for black region is found, this will not affect the weak/strong goldbach conjectures, since the terms appearing there are sometimes large, sometimes zero, and sometimes negative... compared to expected prime gaps in that region of the table.