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Haran
Haran
10:58 AM
Anybody home?
 
 
5 hours later…
Vepir
Vepir
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)
 
Vepir
Vepir
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
 
Vepir
Vepir
5:08 PM
 
Vepir
Vepir
5:47 PM
(this table extends down and right to infinity)
 
Vepir
Vepir
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.
 

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